Let's say Jorge Bergogio says: "I am converting to Buddhism as of right now." He'd be gone as of that very moment and would lose office at that moment. There need be no juridical process to declare. "Jorge Bergoglio has left the Church and is no longer pope."
Let's say that Bergoglio says: "I am converting to Buddhism as of right now." 30 days later the Church gathers in Council and declares that Bergoglio has lost office. So WHEN did loss of office take place. At the moment that Bergoglio made manifest his leaving of the Church or at the time of the declaration 30 days later? Answering that question is key to resolving this debate.
In the hypothetical case you proposed, it seems as though he would lose his office before any action was taken by the Church (although I wouldn’t venture the guess the exact moment). The reason is because the act of leaving the Church by a public profession, and declaring that he is converting to Buddhism, severs every external bond of unity with the visible society of the Church. If he were no longer a member of the visible society of the Church, by his own admission, he would no longer be the head of the visible society of the Church, by his own action. The only thing the Church would do is declare that he was no longer Pope. That being said, if there were a mitigating factor (e.g., if he had been drugged, or forced to make the false profession against his will, etc.), and provided he quickly returned to the Church before the declaration was issued, I believe he would remain pope.
But in the case of a member of the hierarchy who publicly errs in the faith, and is suspected or believed a heretic, yet who continues to profess being a Catholic and remains united to the visible society of the Church, he will not lose his office unless the Church legally established that he’s a heretic and deposed him, or (in the case of a Pope) declares him deposed.
Bellarmine teaches that those are the two ways a heretical bishop or pope will lose his office. In the case of a Pope, he loses his office is ipso facto, (since the Church has no authority to depose him), but if he has not publicly separate himself from the visible society of the Church, he only loses his jurisdiction when the Church has determined that he’s a heretic and, as far as we are concerned, legitimately declared him deposed.
As in the case of a heretical bishop who, as you conceded, retains his office and jurisdiction as long as he remains externally united to the Church, and is being tolerated by the Pope, so likewise will a heretical pope will retain his jurisdiction as long as he remains united to the visible society of the Church, and is being tolerated by her.
They key difference is whether he continues to be a Catholic by external profession (not a Buddhist by external profession), and remains externally united to the visible society of the Church (not openly leaving the Church, as in the hypothesis you proposed). Bellarmine discusses this at lengthy in De Ecclesia Militante, and goes so far in defending it that he even teaches that manifest heretics, Jews and Pagans are “members of the Body of the Church” if they meet these two conditions.
I think Bellarmine went too far, since baptism is necessary for true membership, but his error helps us to understand his position on what is necessary for someone to become separated from the Church. Listen to what he wrote in response to the teaching of Pope of Pope Nicholas, who said “the Church is the gathering of Catholics”:
“We are necessarily compelled to say that they are called Catholics who profess the Catholic Faith, irrespective of their internal faith. (…) Moreover, it is clear that a gathering of Catholics cannot be formed other than by convoking into one place all those who are said to be and are called Catholic, that is, those who publicly profess themselves to be Catholic.” (De Ecclesia Militante, bk III, ch. X)
According to Bellarmine, if person profess himself to be Catholic, and remain visibly united with the other Catholics, he is a member of the body of the Church. If a professing Catholic is a heretic, he only ceased to be a member, according to Bellarmine, when he breaks into open schism and leaves the Church, or is expelled from her.
Listen to what Bellarmine wrote about those who deny the Faith, but profess themselves to be Catholic for some temporal benefit. After remarking the Calvin and some others believed such people are not members of the Church, he writes:
“We, however, follow the manner of speaking of the majority [of Catholic theologians], who teach that those who are joined with the other faithful by a purely external profession, are true parts, and therefore members of the Body of the church, although withered and dead. And, first of all, this opinion can be demonstrated by these words of John: “And now many have become Antichrists; they went out from us, but they were not of us; for, if they had been of us, they certainly would have remained with us” (1 John 2). For, in this place, John is speaking of heretics, whom he calls Antichrists, and says that before they went out from us they were not of us—that is to say, they were not Catholics in mind and will, even though they were by external profession; but, after they betrayed themselves and burst forth into open schism, they ceased to be ‘of us’ in any sense. (…)” (De Ecclesia Militante, bk III, ch. X)
Next he explains how a heretical bishop or pope loses his office:
“Secondly, the same is demonstrated by the testimonies of those Fathers who teach, with a general consensus, that those who are outside of the Church have no authority or jurisdiction over the Church. And, indeed, reason itself manifestly teaches the same thing: for how is it at all possible to suppose or imagine that he should have jurisdiction, and therefore be head of the Church, who is not a member of the Church [has openly left the Church]? Who ever heard of a heard, who was not a member? But it is certain (whatever one or another may think) that an occult heretic, if he be a bishop or even the supreme Pontiff, does not lose his jurisdiction, or dignity, or the title of head in the Church, until either he publicly separates himself from the Church, or, being convicted of heresy, is unwillingly separated.” (De Ecclesia Militante, bk III, ch. X)
There’s the two ways Bellarmine believes a pope loses his office for manifest heresy: he either publicly separates himself from the Church, or he is determined to be a heretic by the Church, and legally declared to be separated from it.
The latter is what he believed happened in the case of Pope Liberius. Pay close attention to what he wrote about how Liberius lost the pontifical dignity:
Bellarmine: “Then two years later came the lapse of Liberius, of which we have spoken above. Then indeed the Roman clergy, stripping Liberius of his pontifical dignity, went over to Felix, whom they knew to be a Catholic. From that time, Felix began to be the true Pontiff. For although Liberius was not a heretic, nevertheless he was considered one, on account of the peace he made with the Arians, and by that presumption the pontificate could rightly be taken from him: for men are not bound, or able to read hearts; but when they see that someone is a heretic by his external works, they judge him to be a heretic pure and simple, and condemn him as a heretic.” (De Romano Pontifice, bk IV)
Bellarmine doesn’t say Liberius automatically lost the pontificate without the priests of Rome being involved. Quite the contrary, he says the priests of Rome stripped Liberius of the pontifical dignity, which he says they were permitted to do because they believed him to be a heretic. If you consult the Latin, what he actually says is the priests of Rome “abrogated” Liberius’ Pontificate – that is, they legally rendered the papacy null – even though he did not believe Liberius was guilty of the sin of heresy.
That’s how Bellarmine believes the ipso facto loss of office takes place for a heretical pope who has not publicly left the Church. To be clear, the Church does not authoritatively depose him, but simply determines, to the best of its ability, if the Pope has fallen into heresy. If so, and even the bishops are dead wrong, Bellarmine says “by that presumption the pontificate could rightly be taken from him”.
How can they do so, given the fact that the Church has no authority to depose the Pope? They can do so, according to Bellarmine, because manifest heretics are ipso facto deposed, and therefore they don’t need to be authoritatively deposed. The Church simply determines if he’s a heretic, and declares that he is no longer pope. No need to actually depose him. And the Church has the right to make that determination, even though it does not have the authoritatively judge the Pope, in the true sense of the word (as Bellarmine also explains elsewhere), until he has already been determined to have lost his office – only then can he be judged and punished by the Church.
Neither John XXIII, Paul VI, JP II, Benedict, or Francis publicly left the Church. Nor were any of them “stripped of the Pontifical dignity” by the Church. Therefore, they all retained their office and jurisdiction until death, and Francis retains it until now – no less than any other heretical bishop who is being tolerated by the Church and left in office.