Fr Chazal's position could hold water, but then it's just an R&R thing anyway. I'd say yes to it, but to say the pope is pope, but not really pope, is not Catholic teaching.  Who cares anyway? We've always been subject to heretical authorities. The people can request removal of a heretic bishop, perhaps even the pope, but then, the pope is specifically not subject to deposition, especially when a huge majority accept him as pope. The one thing I know is that no group or person has a satisfactorily complete handle on any of this.  I tend to think the pope is the pope and I think he will be punished for all he's done, to include being declared an anti pope for his heresy.  Just not by the laity. We don't have to accept sin as Bergoglio permits or promotes, but must resist his attempts to dismantle the faith where and when we can, since he's still the pope (imho). We have to combat his garbage with charity while avoiding divisive rhetoric and dogmatic propositions about his state within the church because it isn't going to make him go away.  The best way to combat a guy like him is through prayer and sacrifice unless your position demands more.  The one thing I can't stand is when people demand you believe what they believe or they try to excommunicate you. 

Meg, do you hold that Jorge Bergoglio is a public manifest formal heretic?

Moral theology does require you to make that decision.

"The very act of submission to the pretended authority of an openly heretical enemy of the Catholic faith constitutes per se an objectively grave act not only of indiscreet obedience; but done in ignorance, constitutes an act of material schism as well. Thus, while the Recognize and Resist policy of Catholics towards the errant conciliar popes was morally justified from the time of the post-council up to the end of February 2013, when Pope Benedict went into what is increasingly seen to be a coerced retirement; it is no longer morally licit to adhere to it for so long as the heretical intruder (or another like him) remains in power, because it is morally wrong and schismatic to recognize and be subject to a manifestly formal heretic."

Kramer, Paul. To deceive the elect: The catholic doctrine on the question of a heretical Pope . Kindle Edition.

Meg, do you submit to Vatican II and the New Rite of Mass?

Meg, does the Catholic Church require you to make the decision to reject Vatican II and the New Mass?

The fourth opinion is of Cajetan [322]. There, he teaches, that a manifestly heretical Pope is not ipso facto deposed; but can and ought to be deposed by the Church. Now in my judgment, such an opinion cannot be defended. For in the first place, that a manifest heretic would be ipso facto deposed, is proven from authority and reason. The Authority is of St. Paul, who commands Titus [323], that after two censures, that is, after he appears manifestly pertinacious, an heretic is to be shunned: and he understands this before excommunication and sentence of a judge. Jerome comments on the same place, saying that other sinners, through a judgment of excommunication are excluded from the Church; heretics, however, leave by themselves and are cut from the body of Christ, but a Pope who remains the Pope cannot be shunned. How will we shun our Head? How will we recede from a member to whom we are joined?

Now in regard to reason this is indeed very certain. A non-Christian cannot in any way be Pope, as Cajetan affirms in the same book [324], and the reason is because he cannot be the head of that which he is not a member, and he is not a member of the Church who is not a Christian. But a manifest heretic is not a Christian, as St. Cyprian and many other Fathers clearly teach [325]. Therefore, a manifest heretic cannot be Pope. Cajetan responds in a defense of the aforementioned treatise, chapter 25, and in the treatise itself chapter 22, that a heretic is not a Christian simply; but is relatively. For since two things make a Christian, faith and the character, a heretic loses the virtue of faith, but still retains the character; and for that reason, still adheres in some way to the Church, and has the capacity for jurisdiction: hence, he is still Pope, but must be deposed, because he has been disposed due to heresy; after being disposed at the last, he is not Pope, as such he is a man, and not yet dead, but constituted at the point of death.

But on the contrary, since in the first place, were a heretic to remain joined with the Church in act by reason of the character, he could never be cut off and separated from her, because the character is indelible, yet everyone affirms that some can be cut off from the Church de facto: therefore, the character does not make a heretical man exist in the Church in act; rather, it is only a sign that he was in the Church, and that he ought to be in the Church. Just as the character impressed upon a sheep, when it was in the mountains, does not make it to be in the sheepfold, rather indicates from which fold it fled, and to where it can be driven back again. This is also confirmed by St. Thomas [326], who says that those who do not have faith are not united to Christ in act, but only in potency, and there he speaks on internal union, not external, which is made through the confession of faith, and the visible Sacraments. Therefore, since the character pertains to what is internal and not external, according to St. Thomas, the character alone does not unite a man with Christ in act. Next, either faith is a necessary disposition as one for this purpose, that someone should be Pope, or it is merely that he be a good Pope. If the first, therefore, after that disposition has been abolished through its opposite, which is heresy, and soon after the Pope ceases to be Pope: for the form cannot be preserved without its necessary dispositions. If the second, then a Pope cannot be deposed on account of heresy. On the other hand, in general, he ought to be deposed even on account of ignorance and wickedness, and other dispositions which are necessary to be a good Pope, and besides, Cajetan affirms that the Pope cannot be deposed from a defect of dispositions that are not necessary as one, but merely necessary for one to be a good Pope.

Cajetan responds that faith is a necessary disposition simply, but in part not in total, and hence with faith being absent the Pope still remains Pope, on account of another part of the disposition which is called the character, and that still remains. But on the other hand, either the total disposition which is the character and faith, is necessary as one unit, or it is not, and a partial disposition suffices. If the first, then without faith, the necessary disposition does not remain any longer as one, because the whole was necessary as one unit and now it is no longer total. If the second, then faith is not required to be good, and hence on account of his defect, a Pope cannot be deposed. Thereupon, those things which have the final disposition to ruin, soon after cease to exist, without another external force, as is clear; therefore, even a heretical Pope, without any disposition ceases to be Pope through himself.

Next, the Holy Fathers teach in unison, that not only are heretics outside the Church, but they even lack all Ecclesiastical jurisdiction and dignity ipso facto. Cyprian says: “We say that all heretics and schismatics have not power and right” [327]. He also teaches that heretics returning to the Church must be received as laymen; even if beforehand they were priests or bishops in the Church [328]. Optatus teaches that heretics and schismatics cannot hold the keys of the kingdom of heaven, nor loose or bind [329]. Ambrose and Augustine teach the same, as does St. Jerome who says: “Bishops who were heretics cannot continue to be so; rather let them be constituted such who were received that were not heretics” [330]. Pope Celestine I, in an epistle to John of Antioch, which is contained in Volume One of the Council of Ephesus, ch. 19, says: “If anyone who was either excommunicated or exiled by Bishop Nestorius, or any that followed him, from such a time as he began to preach such things, whether they be from the dignity of a bishop or clergy, it is manifest that he has endured and endures in our communion, nor do we judge him outside, because he could not remove anyone by a sentence, who himself had already shown that he must be removed.” And in a letter to the clergy of Constantinople: “The Authority of our See has sanctioned, that the bishop, cleric or Christian by simple profession who had been deposed or excommunicated by Nestorius or his followers, after the latter began to preach heresy, shall not be considered deposed or excommunicated. For he who had defected from the faith with such preaching, cannot depose or remove anyone whatsoever.”

Nicholas I confirms and repeats the same thing in his epistle to the Emperor Michael. Next, even St. Thomas teaches that schismatics soon loose all jurisdiction; and if they try to do something from jurisdiction, it is useless [331]. Nor does the response which some make avail, that these Fathers speak according to ancient laws, but now since the decree of the Council of Constance they do not lose jurisdiction, unless excommunicated by name, or if they strike clerics. I say this avails to nothing. For those Fathers, when they say that heretics lose jurisdiction, do not allege any human laws which maybe did not exist then on this matter; rather, they argued from the nature of heresy. Moreover, the Council of Constance does not speak except on the excommunicates, that is, on these who lose jurisdiction through a judgment of the Church. Yet heretics are outside the Church, even before excommunication, and deprived of all jurisdiction, for they are condemned by their own judgment, as the Apostle teaches to Titus; that is, they are cut from the body of the Church without excommunication, as Jerome expresses it.

Next, what Cajetan says in the second place, that a heretical Pope who is truly Pope can be deposed by the Church, and from its authority seems no less false than the first. For, if the Church deposes a Pope against his will, certainly it is over the Pope. Yet the same Cajetan defends the opposite in the very same treatise. But he answers; the Church, in the very matter, when it deposes the Pope, does not have authority over the Pope, but only on that union of the person with the pontificate. As the Church can join the pontificate to such a person, and still it is not said on that account to be above the Pontiff; so it can separate the pontificate from such a person in the case of heresy, and still it will not be said to be above the Pope.

On the other hand, from the very fact that the Pope deposes bishops, they deduce that the Pope is above all bishops, and still the Pope deposing a bishop does not destroy the Episcopacy; but only separates it from that person. Secondly, for one to be deposed from the pontificate against his will is without a doubt a penalty; therefore, the Church deposing a Pope against his will, without a doubt punished him; but to punish is for a superior and a judge. Thirdly, because according to Cajetan and the other Thomists, in reality they are the same, the whole and the parts are taken up together. Therefore, he who has so great an authority over the parts taken up together, such that he can also separate them, also has it over the whole, which arises from those parts.

Furthermore, the example of Cajetan does not avail on electors, who have the power of applying the pontificate to a certain person, and still does not have power over the Pope. For while a thing is made, the action is exercised over the matter of the thing that is going to be, not over a composite which does not yet exist, but while a thing is destroyed, the action is exercised over a composite; as is certain from natural things. Therefore, when Cardinals create the Pontiff, they exercise their authority not over the Pontiff, because he does not yet exist; but over the matter, that is, over the person whom they dispose in a certain measure through election, that he might receive the form of the pontificate from God; but if they depose the Pope, they necessarily exercise authority over the composite, that is, over the person provided with pontifical dignity, which is to say, over the Pontiff.

Now the fifth true opinion, is that a Pope who is a manifest heretic, ceases in himself to be Pope and head, just as he ceases in himself to be a Christian and member of the body of the Church: whereby, he can be judged and punished by the Church. This is the opinion of all the ancient Fathers, who teach that manifest heretics soon [mox — better translation: immediately] lose all jurisdiction, and namely St. Cyprian who speaks on Novation, who was a Pope in schism with Cornelius: “He cannot hold the Episcopacy, although he was a bishop first, he fell from the body of his fellow bishops and from the unity of the Church” [332]. There he means that Novation, even if he was a true and legitimate Pope; still would have fallen from the pontificate by himself, if he separated himself from the Church. The same is the opinion of the learned men of our age, as John Driedo teaches [333], those who are cast out as excommunicates, or leave on their own and oppose the Church are separated from it, namely heretics and schismatics. He adds in the same work [334], that no spiritual power remains in them, who have departed from the Church, over those who are in the Church. Melchior Cano teaches the same thing, when he says that heretics are not part of the Church, nor members [335], and he adds in the last Chapter, 12th argument, that someone cannot even be informed in thought, that he should be head and Pope, who is not a member nor a part, and he teaches the same thing in eloquent words, that secret heretics are still in the Church and are parts and members, and that a secretly heretical Pope is still Pope. Others teach the same, whom we cite in Book 1 of de Ecclesia. The foundation of this opinion is that a manifest heretic, is in no way a member of the Church; that is, neither in spirit nor in body, or by internal union nor external. For even wicked Catholics are united and are members, in spirit through faith and in body through the confession of faith, and the participation of the visible Sacraments. Secret heretics are united and are members, but only by an external union: just as on the other hand, good Catechumens are in the Church only by an internal union but not an external one. Manifest heretics by no union, as has been proved.