Odd. Billuart held the opposite and less common opinion to Bellarmine that even occult heretics who had not publicly manifested their heresy were not members of the Church (see cited Fr. Fenton article, page 214 - the opinion of Sylvius, page 211), since they lacked the requisite supernatural bond of faith with Christ.

The opinion, contrary to Bellarmine's, that even occult heretics who have not manifested their heresy publicly are not members of the Church is a permissible opinion on that still open question. Father Fenton talks about it here:

Status_St_Robert_Bellarmine_Membership_Occult_Heretics_Church.pdf (ecclesiamilitans.com)

My own view is that the question is an academic one, since, even if Francis is not a member and not pope, he de facto is pope, sitting in the seat, making cardinals, etc., with no rival claimant out there.

True.

On the Deposition of the Pope (Part 2 of 2) - Dominicans of Avrille, France (dominicansavrille.us)

In the Treatise on the Incarnation (De Incarnatione, diss. IX, a. II, § 2, obj. 2) Billuart defends the thesis that Christ is not the head of heretics, even occult.

It is objected that several doctors (Cajetan, Soto Cano, Suárez, etc.) say that the Pope fallen into occult heresy remains the head of the Church. So he must be a member.

Billuart denies the conclusion:

There is a difference between being constituted a head by the fact that one is influencing on the members, and being made a member by the fact that one is receiving an influx in itself;  this is why, while the pontiff [who] fell into occult heresy keeps the jurisdiction by which he influences the Church by governing her, thereby he remains the head;  but as he no longer receives the vital influx of Christ‘s faith or charity, who is the invisible and first head, he cannot be said to be a member of Christ or of the Church.

Instance: it is repugnant to be the head of a body without being a member, since the head is the primary member.

Answer:  I distinguish the first sentence: it is repugnant to a natural head, I agree; to a moral head, I deny it.  For example, Christ is the moral head of the Church, but he is not a member.  The reason for the difference is that the natural head cannot have an influence on other members without receiving the vital influx of the soul.  But the moral head, as the Pontiff is, can exercise the jurisdiction and the government over the Church and its members, although he is not informed by the soul of the Church, which are faith and charity, and that he does not receive any vital influx. 

In a word, the Pope is made a member of the Church through the personal faith which he can lose, and the head of the Church by the jurisdiction and the power which can be reconciled with an internal heresy. (Cursus theologiœ, Pars III, Venice, 1787, p. 66)
In the Treatise on Faith (De Fide diss IV to III, § 3, obj 2) Billuart defends the following thesis:  Heretics, even manifest (unless being denounced by name, or by leaving the Church themselves) keep the jurisdiction and absolve validly.
He considers the question of the case of a Pope, which is a special case, who receives his jurisdiction not from the Church, but directly from Christ:

It is nowhere stated that Christ continues to give jurisdiction to a manifestly heretical Pontiff, for this can be known by the Church and she can get another pastor.  However the common sentence [editor:  opinion] holds that Christ, by a special provision (ex speciali dispensatione), for the common good and peace of the Church, continues [to give] jurisdiction to a Pontiff even who is a manifest heretic, until he is declared manifestly heretical by the Church. (Cursus theologiœ, Pars II-II, Brescia, 1838, p. 33-34)

In the Treaty on the Rules of Faith (De regulis fidei, diss IV, VIII a, § 2, obj 2 and 6) Billuart defends the following thesis:  The sovereign Pontiff is superior to any council by authority and jurisdiction.

It is objected that the Pontiff is subject to the judgment of the Church in the case of heresy.  Why then he would not be subject also in other cases?

He replies:

This is because in the case of heresy, and not in other cases, he loses the pontificate by the fact itself of his heresy: how could remain head of the Church he who is no longer a member?  This is why he is subject to the judgment of the Church, not in order to be removed, since he is already deposed himself by heresy and he rejected the Pontificate (pontificatum abjecerit), but in order to be declared a heretic, and thus that he will be known to the Church that he is not anymore Pontiff: before this statement [of the Church] it is not permitted to refuse him obedience, because he keeps jurisdiction until then, not by right, as if he were still Pontiff, but in fact, by the will of God and accordingly disposing it for the common good of the Church. (Cursus theologiœ, Pars II-II, Brescia, 1838, p. 123)

Another objector remarked that the Church would be deprived of a remedy if she could not subject the Pope to the Council in the case that he would be harmful and would seek to subvert her.

Billuart replied that:

If the pope sought to harm her in the faith, he would be manifestly heretical, and he would thereby lose the Pontificate: however it should be necessary a declaration of the Church in order to deny him obedience, as we have said above. (Cursus theologiœ, Pars II-II, Brescia, 1838, p. 125)

If the Pope would harm the Church otherwise than in the faith, some say that one could resist him by the force of arms, however without losing his superiority.  St. Thomas Aquinas said it would be necessary to appeal to God in order to correct him or taking him away from this world (4 Sent. D. 19, q. 2, a. 2 q.1a 3, ad 2).

Billuart prefers to think that:

Whereas God governs and sustains his Church with a special Providence, he will not permit, as he has not permitted it so far, that this situation will happen, and if he permits it, he will not fail to give the means and the help appropriate. (Cursus theologiœ, Pars II-II, Brescia, 1838, p. 125)