Are you saying St. Robert Bellarmine misunderstood what he read in the Fathers?

I've read True or False Pope, and they don't say Bellarmine and Cajetan held the same opinion.  On the contrary, they do a very good job explaining the difference between the two opinions.

I will explain how Bellarmine was trying to defend his position, and it is not what most people think.  The first thing to realize is that none of the Fathers taught that "manifest heretics are ipso facto deposed".  That's why Bellarmine was unable to quote any who taught it.

What the Fathers did teach is that 1) heretics (meaning those who were visibly separated from the Church) and also schismatics (who were visibly separated from the Church) were outside of the Church; and 2) because they were outside of the Church they lacked jurisdiction. The reason it was necessary for Bellarmine to establish point #1 is because some theologians in Bellarmine's day held that heretics and schismatics who had publicly left the Church, or were never in the Church, were still part of the Church.  Bellarmine proved that the Fathers taught otherwise.  If they were separated from communion with the Church, they were not part of the Church.

But it is one thing to say a heretic who is outside the Church lacks jurisdiction, and another to say a Catholic bishop who falls into heresy loses jurisdiction.  Not having jurisdiction is one thing; losing it is another.   Joel Osteen is a heretic who lacks jurisdiction, not because he lost it, but because he never had it.

What the quotations that Bellarmine references from St. Athanasius, St. Augustine, Optatus, St. Ambrose, and the two others from St. Cyprian that he quotes confirm, is that those who are outside the Church lack jurisdiction.  And the quotations from St. Cyprian and Ambrose were both referring to Novation, who never had jurisdiction to begin with.  Nothing in any of those quotes speaks of how someone who possessed jurisdiction in the Church would lose it if he fell into heresy and/or left the Church.

What Bellarmine was trying to argue is that since the Fathers all taught that heretics and schismatics (who were visibly separated from the Church) lacked jurisdiction, if a Pope was outside the Church, he too would lack jurisdiction.  The problem, as noted above, is that lacking jurisdiction and losing jurisdiction are two different things.  

The only example Bellarmine used of a bishops who possessed jurisdiction before losing it was Nestorius, but if anything, the case of Nestorius refutes Bellarmine's position.  Nestorius wasn't ipso facto deposed.  He retained his office from the day he first preached his new heresy (Christmas of 428), until he was deposed by the Council of Ephesus (July 431).  Between those two date, Nestorius had been issued to formal warnings by St. Cyril, the Patriarch of Alexandria, he had been judged by the Pope during a council held in Rome (August 430), and had been issued a third and final warning by the Pope via St. Cyril, following the Council, giving him 10 days to renounce his heresies. Up to that point, Nestorius remained "in communion with the Church," according to the Pope himself. And when the Pope learned that the emperor had called a council to consider the matter, he agreed to suspended the sentence of deposition that he said would take effect after the 10 days were up, and allowed Nestorius to remain in is see until the Council rendered a judgement.   Only then was Nestorius deposed.

That is the only example Bellarmine referenced of a bishop who lost his jurisdiction.   And the only aspect of the story Bellarmine related is that after the Pope judged Nestorius at the Council of Rome (August 430), he wrote to those who had been excommunicate by Nestorius to let them know the excommunications that had been pronounced against them were null.    

To say the least, Bellarmine completely failed to prove his position.  He didn't provide one convincing argument in favor of it or quote any authority who supported it.  Again, if you disagree, point out anything Bellarmine wrote that you find convincing, or any authority he cited that you believe supports his position.

I believe that the Nestorius episode actually backs sedeprivationism, where on account of his manifest heresy, Nestorius lacked authority even before he officially (materially) lost his office.