Here's the bottom line ... thanks to the article linked to by Decem.

We can argue until the cows come home about what Bellarmine meant ... and it doesn't really matter, since this question has not been definitively settled by the Church.

Even St. Robert's opinion regarding the requirements for membership in the Church have not been authoritatively settled.  There's another school of thought, which, although it has become less common these days, which holds, among other things, that occult heretics are not fully members of the Church.  I think that the position has a lot of merit.  And there are a tremendous number of permutations regarding these views.

So, in other words, of someone wanted to hold even that occult heretics would lose the papal office, there's nothing to stop them.

So we're wasting our time debating this issue.

Any theory must abide by things that the Church HAS settled, such as that Councils cannot formally depose popes (Lateran V), and that the Holy See is judged by no one, and that the Pope has supreme absolute authority in the Church (Vatican I).  Any theory that does not contradict these principles can be held by a Catholic.  What we're arguing about is whether some distinction or another is legitimate so that some theory which APPEARS to contradict one of these principles really doesn't.

Except St Robert Bellarmine's teaching was used at Vatican I.  It can not be simply dismissed as just another opinion.

Well, the article in which Msgr. Fenton was discussing Father Lawlor was written in 1950 ... so I would assume so.

According to Msgr. Fenton, there still exists a division between the two major schools, St. Robert and Sylvius, although the former is "more common".

My apologies...I think I found the wrong Fr Lawlor on the internet.  For me, occult heretics do not affect the unity of the Church...which is what I believe is the main driving force for St Robert Bellarmine's teaching on manifest heretic popes.

This is a very complicated issue, not settled yet by the Church, and this 37 pages of argument will lead us to no definitive conclusion.

I feel that we're wasting too much time on arguing about it.

What the REAL issue here is whether the Catholic Magisterium and Universal Discipline of the Church can become THIS corrupt.  R&R says yes, whereas the sedevacantists/privationists say no.  That's the REAL discussion here.  If the Magisterium and Universal Discipline had remained intact, then the heretical ramblings of one Jorge Bergoglio would not matter very much to the average Catholic.

You can find theologians on all sides of the heretical pope issue.  But I defy anyone to find a pre-Vatican II theologian who would say that it is not heretical to hold that an Ecuмenical Council could teach grave error to the Church, leading souls to hell, and that the Church could approve of and promulgate a Mass that's harmful to souls and displeasing to God.  THAT is the real issue here.  This heretical pope issue is a distraction.

I'm just beginning to read this; it is excellent.  I have not yet gotten to his conclusion, but he makes some great points regarding the state of the question.

Here's an interesting corollary to Bellarmine's purely-visible concept of the Church, one that has always troubled me.  St. Robert declares that excommunicates are outside the Church.  But what about those unjustly excommunicated?  Although visibly separated from the body of the Church, they are in fact still part of the Church because the excommunication was unjust.

Q. Stubborn asks: "Specifically, what member or members of said Church are supposed to make this an official declaration?"

A. How many times do you expect me to repeat myself? Go back and read my earlier comments. I have already answered this question.