Thanks for posting the link.
In the link you provided, where does Bellarmine make the distinction between the "matter" and "form" of the office?
I didn't link you the full thing, apologies. https://novusordowatch.org/de-romano-pontifice-book2-chapter30/
the Pope immediately ceases to be Pope: for the form cannot maintain itself without the necessary dispositions.
The example of the electors, who have the power to designate a certain person for the pontificate, without however having power over the Pope, given by Cajetan, is also destitute of value. For when something is being made, the action is exercised over the matter of the future thing, and not over the composite, which does not yet exist, but when a thing is destroyed, the action is exercised over the composite, as becomes patent on consideration of the things of nature. Therefore, on creating the Pontiff, the Cardinals do not exercise their authority over the Pontiff for he does not yet exist, but over the matter, that is, over the person who by the election becomes disposed to receive the pontificate from God. But if they deposed the Pontiff, they would necessarily exercise authority over the composite, that is, over the person endowed with the pontifical power, that is, over the Pontiff.
Now the fifth true opinion, is that a Pope who is a manifest heretic, ceases in himself to be Pope and head, just as he ceases in himself to be a Christian and member of the body of the Church: whereby, he can be judged and punished by the Church.
The pope cannot be deposed, but the man still clinging onto the crown after he deposed himself through formal heresy can be deposed. The cardinals can act upon the matter and elect the pope, and if the election is valid then the pope receives the form from God. If a pope becomes a manifest heretic, he loses the form and no longer holds the spiritual office - and therefore he can be deposed. But the matter will still be sitting there on the throne of St. Peter until he's dragged off it, and he might be an occult heretic for many years before he's outed as a manifest heretic. So the Church provides and revokes the matter, but God provides and revokes the form.
Now it's easy to read this as "the pope is not the pope from the instant he embraces heresy, and the deposition is just a formality to let the faithful know". But if you look at the second opinion he addresses, which is exactly that, he contradicts this view.
Thus, the second opinion is that the Pope, in the very instant in which he falls into heresy, even if it is only interior, is outside the Church and deposed by God, for which reason he can be judged by the Church. That is, he is declared deposed by divine law, and deposed de facto, if he still refused to yield. This is of John de Turrecremata , but it is not proven to me. For Jurisdiction is certainly given to the Pontiff by God, but with the agreement of men, as is obvious; because this man, who beforehand was not Pope, has from men that he would begin to be Pope, therefore, he is not removed by God unless it is through men. But a secret heretic cannot be judged by men, nor would such wish to relinquish that power by his own will. Add, that the foundation of this opinion is that secret heretics are outside the Church, which is false, and we will amply demonstrate this in our tract de Ecclesia, bk 1.
It's clear from this paragraph that the Church deposition is not a mere formality and that the man is still pope until the Church declares him a manifest heretic and deposes him.