Rather:
Popes must be Catholic.
As a manifest, pertinacious heretic, Bergoglio is not Catholic.
Therefore, Bergoglio is not pope.
No judgement of any council is required, nor possible, as a pope is judged by no man or council of men. Manifest, pertinacious heretics automatically fall from office. This is the way they lose their office, since no man can judge them.
Unfortunately, it's not quite this simple. While I lean in this direction, R&R would reject and/or qualify your minor. They would perhaps argue that pertinacity is established in the external forum only after he remains pertinacious after formal admonition from the Church. Who is competent to make a finding of pertinacity? Fr. Cekada's "Aunt Helen"? Now, it's much easier with Bergoglio, but it would have been much more difficult to establish pertinacity on the part of Ratzinger.
With regard to the sentence beneath your syllogism, Cajetan, John of St. Thomas, and others distinguish that. While they agree that no Pope can be judged juridically, the Church must intervene and make a "finding of fact" judgment against Bergoglio before he would be stripped of his office. Then there's the material vs. formal distinction, that Bergoglio would remain in material possession of the office until the Church declares otherwise.
Actually, the principles of sedeprivationism (which I strongly favor) reconcile the Bellarmine and Cajetan/John of St. Thomas positions quite nicely, and the quote adduced by St. Robert Bellarmine from St Celestine regarding the case of Nestorius would actually back up sedeprivationism.