What's lost in all this is that Bellarmine did not hold any of these opinions about the Pope. He personally held that a legitimate Pope would never be allowed by God to fall from office.
Yes, he held this position of Pighius that the Pope could not fall into heresy because, as he said, it seemed to be in accordance with the sweet dispositions of Divine Providence (or very similar words that I can't put my hand on right now). But he acknowledged that this was not the common opinion, which is why he examined the subject further.
Bishop Vincent Gasser in his Official
Relatio for the
Deputatio de Fide at the First Vatican Council explained, quashing rumours, that the Council had no intention whatsoever of confirming this "extreme opinion", nor any other extreme opinion. It is worth reading.