This got my attention. Although I don't currently agree with it, it's an intriguing question which I am open to accept if sufficient evidence is found. Now when I read classical works on the Papacy (or general catholic works on Ecclesiology), I'm starting to see sort of how St. Robert Bellarmine came to that conclusion. Some of the "evidence" I've found so far are Saurez's Defensio Fidei Catholicae (taken as a whole) and Sermons by St's. Gregory the Great and Leo the Great.
"When the Passion of the Lord was drawing near, an event that was going to shake the constancy of his disciples, he said: 'Simon, Simon. Behold, Satan has obtained his request to sift you (all) like wheat. I, however, have begged for you that your faith not fail. Once you have converted, strengthen your brethren, lest you (all) enter into temptation.' Each apostle encountered the same danger through temptation from fear. All equally needed the help of divine protection, since the devil wanted to harass them all and to crush them all. Still, the Lord took special care of Peter and prayed especially for Peter. It was as if the condition of the others would be more secure if the mind of their leader were not overcome. In Peter, therefore, the fortitude of all is reinforced, for the aid of divine grace is ordered in such a way that the firmness given to Peter through Christ is conferred upon the apostles through Peter."" (Tractatus IV, Pars 3, Sancti Leonis Magni).
Now, I'm not arguing that that quote alone proves the thesis but combined with other arguments in context may give some weight. Does anyone have any evidence for St. Robert Bellarmine's claim? Any arguments against it?