The way I understand it is that sedevacantists do not believe the church can depose a pope because the first see is judged by no-one.
They say that the Pope spouting public heresy would in and of itself sever him from the papacy and that the cardinals would simply declare the see vacant for the purpose of electing a new pope.

Putting the pope on trial for heresy would be a practical impossibility. He could simply dismiss everyone attempting to put him on trial or refuse to participate as he is the supreme authority.

But the vast majority of sedevacantists do not argue this. They instead argue that most of the the Vatican II papal claimants were public heretics before their election and thus could not become true popes under divine law.

Wrong.  The vast majority of sedevacantists believe that the VII popes are false because they promulgated (Paul VI), profess and teach a false religion to the Universal Church.

I wrote the webmaster at Fr. Wathen’s official website. Here is what he said. 

“Yes, I would agree that Fr Wathen (as you described):

“recognized the Vatican II popes as true popes, but resisted their commands that ran contrary to Tradition.”

As to if Fr was a sedeprivationist, I would say that is hard to say because the term is newly coined, as the wiki article you linked described.  So did Fr even know that this term existed?  I don’t know.  

All I can say is that Fr did call the post-conciliar popes heretics and revolutionaries and he did not mince words here.  I also know he was opposed to sedevacantism because it was a distraction from one's daily catholic duties.  

Did Fr view the papal question like the sspx, ie Recognize and Resist?  Yes, partially.  I say partially because the sspx has always been reluctant to call the post-conciliar pope’s actions as heretical.  Did Fr also say that post-conciliar pope’s authority is limited?  Yes, because he said that if any of the post-conciliar pope’s converted, they could get rid of their “censures” by confession.  A censure is an ecclesiastical penalty for spiritual sins, which includes heresy.  A pope (or anyone) who is a private heretic incurs ecclesiastical penalties even if no one else would know, except God.  

I don’t think Fr spent the time to adequately explain his views (because he didn’t think it was the best use of his time) but I do think he recognized that a heretical pope incurs spiritual penalties (which is similar to sedeprivationism in some aspects.)

I’m sorry I couldn’t help you on your question.  Have a good day and God bless,

Thomas”

Can a person be re-baptized?  Just answer the question.
If no, then you have to admit that there are different meanings and understandings to the word "membership".

Also, the term "imperfect communion" is a modernistic novelty.  So who cares if heretics are "imperfectly" united to the Church?  They're still going to hell.  "Imperfect communion" is theologically meaningless and only used because V2 wants to support their humanistic agenda and a one-world religion.  
There is a distinction between belief and the indelible mark.  No matter what a man does in this life, he cannot erase the indelible mark, and if he goes to hell, the devils will torment him much more than an unbaptized person because a baptized person was given the ultimate grace of the Faith, which he rejected.

No. He cannot.

However, although heretics retain the character of Baptism and Priesthood (see Trent), they are not members or parts of the Church. They are excluded. The opinion that protestants and schismatics are still considered members of the Church even though they profess false doctrines, is clearly false and can easily be refuted.

Bellarmine touches this very point, on the Church Militant, Chapter V, on Heretics and Apostates:

Then, obviously, there must be different understandings of the word 'membership', right?

1.  Unbaptized, non-catholics.
2.  Baptized, non-catholics.
3.  Baptized catholics.

If you apply the word 'membership' equally to the 3 classes of people above, this is incorrect.  Membership means different things in different scenarios.  That's my only point.

Membership only applies to the third category.

Unbaptized non-Catholics and Baptized non-Catholics (heretics, apostates, and schismatics) are not members.

Why is this important in the context of this thread is that a non-member simply cannot be the head of the Church Militant since he is not even a part of it. He loses Authority over the members therein.

Yes, in a sense, they are still members.  If they lost their membership 100%, then they would have to be re-baptized.

Not according to Msgr. Fenton ... who doesn't believe in the concept of an imperfect membership.  I actually believe that it's possible, and could be used as a Catholic explanation for BoD.  I had a big argument with LoT about this very subject.  Fenton rejected the notion of imperfect membership held by some theologians, but I felt that he was wrong.  At least with that you don't have to come up with an ecclesiology that allows the salvation of catechumens that results in this ridiculous idea of an invisible union with a visible body.  He says that you can be within the Church without being part of it ... with the only visualization being not unlike an undigested piece of food inside the stomach.  So I call that "undigested hamburger ecclesiology".

BUT ... generally speaking, according to St. Robert Bellarmine and others, heretic, apostates, and schismatics are NOT members of the Church.  It's not unlike if someone cut off your hand.  By virtue of the DNA, it still retains the identifying marker of having once been part of the body, but it's no longer part of the body.  Similarly, the baptismal character of such a one indicates that the person USED to be part of the body, even though it no longer is (as taught by St. Thomas).  So just like with the DNA markers of a severed limb.  It's not just a dead member, but has actually been removed from the unity of the body.  Now, if part of your body rots of gangrene or something, it's a dead member, but still a part of the body.  Hair and nails, for instance, consist of dead cells, but they are part of the body.  That's like Catholics who are not in a state of grace.

The word "privationism" probably wasn't even coined back when Fr. Wathen wrote the book, but in the specified paragraph in the OP he seems to posit some of the concept of privationism.

You can compare this to the idea of sedeprivationism.  A heretic is a passive but not active member of the Church, in a manner of speaking.

Not so fast.

There is a difference between a formal heretic and material heretic. A formal heretic is outside of the Church. He isn't part of the Church Body one iota. He is a formal heretic, manifestly, by his obstinate rejection of any Catholic dogma & doctrine, even if the Church hierarchy hasn't formally excommunicated him. He is ipso facto excommunicated when he is shown his heresy so he will publicly abjure and repent, yet, he willingly adheres to and promulgates that heresy or heresies.

A material heretic is a Catholic erring in good faith, but he can still a member of the Church, if he hasn't been shown his heresy (so he can correct himself & know the Truth) regarding defined/affirmed Dogma & doctrine of the Catholic Faith or any part of the Catholic Faith given to us through Holy Tradition's word and/or epistle.

Just as the faithless Jєωs, who were chosen by God to hold, and be a living example of, the only true Faith - the only conduit to the only true God, and to be a light to the Gentiles,... these same Jєωs repetitively broke their covenants with God and, finally, in rejecting Jesus Christ - the everlasting Covenant Who dwelt among them so they could see their continual errors, repent and be saved - they ruptured themselves from the Bosom of Abraham, thus being damned eternally... likewise, any Catholic who obstinately rejects defined/affirmed dogma & doctrine of the Catholic Faith, he rejects Jesus Christ Himself, thus he ceases being a member of Christ's Bride - the Church - and he is damned eternally.

I can't find the exact St. Thomas quote, cited (but not quoted) by St. Robert Bellarmine, but he states that the character in a severed member indicates not that the member is still part of the body but merely that it USED to be and has a potency to return thereto.

The citation is III, Q.8, art. 3

Summa Theologica:

I answer that, This is the difference between the natural body of man and the Church's mystical body, that the members of the natural body are all together, and the members of the mystical are not all together---neither as regards their natural being, since the body of the Church is made up of the men who have been from the beginning of the world until its end---nor as regards their supernatural being, since, of those who are at any one time, some there are who are without grace, yet will afterwards obtain it, and some have it already. We must therefore consider the members of the mystical body not only as they are in act, but as they are in potentiality. Nevertheless, some are in potentiality who will never be reduced to act, and some are reduced at some time to act; and this according to the triple class, of which the first is by faith, the second by the charity of this life, the third by the fruition of the life to come. Hence we must say that if we take the whole time of the world in general, Christ is the Head of all men, but diversely. For, first and principally, He is the Head of such as are united to Him by glory; secondly, of those who are actually united to Him by charity; thirdly, of those who are actually united to Him by faith; fourthly, of those who are united to Him merely in potentiality, which is not yet reduced to act, yet will be reduced to act according to Divine predestination; fifthly, of those who are united to Him in potentiality, which will never be reduced to act; such are those men existing in the world, who are not predestined, who, however, on their departure from this world, wholly cease to be members of Christ, as being no longer in potentiality to be united to Christ.

The damned cease to be members of Christ in both, act and potency, even if they were baptized earlier in life. The indelible mark is not sufficient to remain a member of the Church, since the link is cut by heresy, apostasy, or shiscm.