the Pope needs to have supernatural faith... Supernatural faith is not just “necessary for” but a necessary CONSEQUENCE of membership of the Church, which is to say, it’s the very essence thereof

and supernatural faith will result in the external profession of AT LEAST faith in the Trinity and Incarnation, and everything else implicitly, if not explicitly due to ignorance.

Mere external profession of faith does not a member of the Church make, which is why even an occult  heretic cannot be Pope, which is why the Pope NECSSARILY has the gift of unfailing (personal) faith.
As I have stated already (and is persistently ignored by “RomanTheo”, or he wouldn’t argue as he is arguing), Bellarmine’s argument is a reductio ad absurdum: YES, if a pope could become a heretic, then a LOGICAL DEDUCTION shows that he could NOT lose office merely for being an occult heretic;  but this contradicts the fact that personal supernatural faith is essential to membership of the Church.

Using the Principle of Non-Contradiction, Catholics are able to judge that a contradiction does exist, then they can refuse to continue listening to the false pastor. If the false pastor has not been deposed by competent authority, he still occupies his office materially in the meantime until such deposition occurs; however, he has lost all Authority on account of his heretical teaching. This is foundational in the Thesis of Cassisiacuм.

I don't get this. Are talking about the Church physically located in Rome? The physical Hierarchy located in Rome? The inability of the Roman Pontiff to teach error?
This is fairly vague.

3. Therefore whoever succeeds to the chair of Peter obtains by the institution of Christ himself, the primacy of Peter over the whole Church. So what the truth has ordained stands firm, and blessed Peter perseveres in the rock-like strength he was granted, and does not abandon that guidance of the Church which he once received [47].

4. For this reason it has always been necessary for every Church--that is to say the faithful throughout the world--to be in agreement with the Roman Church because of its more effective leadership. In consequence of being joined, as members to head, with that see, from which the rights of sacred communion flow to all, they will grow together into the structure of a single body [48].

III. St. Peter's work is still carried out by his successors

The dispensation of Truth therefore abides, and the blessed Peter persevering in the strength of the Rock, which he has received, has not abandoned the helm of the Church, which he undertook. For he was ordained before the rest in such a way that from his being called the Rock, from his being pronounced the Foundation, from his being constituted the Doorkeeper of the kingdom of heaven, from his being set as the Umpire to bind and to loose, whose judgments shall retain their validity in heaven, from all these mystical titles we might know the nature of his association with Christ. And still today he more fully and effectually performs what is entrusted to him, and carries out every part of his duty and charge in Him and with Him, through Whom he has been glorified. And so if anything is rightly done and rightly decreed by us, if anything is won from the mercy of God by our daily supplications, it is of his work and merits whose power lives and whose authority prevails in his See. For this, dearly-beloved, was gained by that confession, which, inspired in the Apostle's heart by God the Father, transcended all the uncertainty of human opinions, and was endued with the firmness of a rock, which no assaults could shake. For throughout the Church Peter daily says, You are the Christ, the Son of the living God, and every tongue which confesses the Lord, accepts the instruction his voice conveys. This Faith conquers the devil, and breaks the bonds of his prisoners. It uproots us from this earth and plants us in heaven, and the gates of Hades cannot prevail against it. For with such solidity is it endued by God that the depravity of heretics cannot mar it nor the unbelief of the heathen overcome it.

2. Since, however, it would be very tedious, in such a volume as this, to reckon up the successions of all the Churches, we do put to confusion all those who, in whatever manner, whether by an evil self-pleasing, by vainglory, or by blindness and perverse opinion, assemble in unauthorized meetings; [we do this, I say,] by indicating that tradition derived from the apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul; as also [by pointing out] the faith preached to men, which comes down to our time by means of the successions of the bishops. For it is a matter of necessity that every Church should agree with this Church, on account of its preeminent authority [potiorem principalitatem].

don't tell me that a legitimate Ecuмenical Council taught errors (or even heresies) to the Universal Church and don't tell me that a legitimate Pope promulgated a Rite of Mass that displeases God and is harmful to faith.  Don't tell me that it's OK for Catholics to go so far as to break communion with the legitimate hierarchy in order to reject this Magisterium and Universal Discipline.  Don't tell me that the Magisterium and Universal Discipline of the Church have gotten so corrupt that we're almost required to break communion with the Church in order to save our souls.  That is nothing other than the defection of the Catholic Church.