The argument from authority is based on Saint Paul, who orders that the heretic be avoided after two warnings

There's no strict requirement for the two warnings, which is why Bellarmine adds, "that is, after showing himself to be manifestly obstinate".  Two warnings is a rule of thumb for demonstrating obstinacy ... but it's not some kind of divine law condition.  Obstinacy can be manifest in other ways.

How does +Bellarmine define "manifest heresy"?  How is manifest heresy determined?  In his 4th opinion, he says:
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“The fourth opinion is that of Cajetan, for whom the manifestly heretical Pope is not “ipso facto” deposed, but can and must be deposed by the Church. To my judgment, this opinion cannot be defended. For, in the first place, it is proven with arguments from authority, and from reason, that the manifest heretic is “ipso facto” deposed. The argument from authority is based on Saint Paul, who orders that the heretic be avoided after two warnings, that is, after showing himself to be manifestly obstinate – which means before any excommunication or judicial sentence”.   -- De Romano Pontifice, Bk. 2
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+Bellarmine answers both these questions together, when he says that one is "manifestly obstinate" after "two warnings".  +Bellarmine says that this argument is "from authority" meaning it's Scriptural.
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Have any of the V2 popes been rebuked with 2 warnings?  No.  Can we say that they are "manifestly obstinate"?  No.  Have they then lost their office "ipso facto"?  No.
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Can this be any clearer?  

There's plenty wrong with it, starting with the fact that S&S refused to cite all the Canonists who unanimously reject their interpretation (which they appear to have pulled out of thin air).
https://stevensperay.wordpress.com/2016/02/19/the-sin-of-heresy-why-john-salza-and-robert-siscoe-get-it-wrong-part-ii/

You're mixing and matching 2 different problems with 2 different solutions.  As we both agree, there is the material and the spiritual aspect of the papacy.  Or, there are human and the divine elements of the papacy.  When Pius XII is talking about heresy severing one from the Body of Christ, he is speaking of the sin of heresy and of the spiritual office.  It's fairly obvious to all of us that one can be guilty of excommunication (spiritually speaking) long before they would be deposed from office (materially speaking).
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In his 4th opinion, when +Bellarmine says that a "manifest" heretic loses office immediately, he is speaking of the material office, which is the whole point of his debates.  When +Bellarmine says that "2 rebukes" are necessary for manifest heresy to be proven, he is arguing that for the MATERIAL OFFICE to be lost, you must have a MATERIAL (i.e. human) proof or process.  The rebukes/warnings are an EXTERNAL manifestation of the heretic's INTERNAL/SPIRITUAL error.
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A heretic, whether formal or material (or occult/private), can still be severed from the Church SPIRITUALLY, which is what Pius XII rightly taught (and what +Chazal says).  But this SPIRITUAL penalty does not affect his MATERIAL OFFICE, because, (from common sense) the internal forum cannot be known or be proven.  So how does the Church depose (i.e. materially remove) someone from office?  There has to be a process, which involves a HUMAN ELEMENT of the GOVERNING aspect of the Church.  This process is St Paul's "2 warnings".
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Thus, as soon as a pope were to speak heresy, they could be presumed to be spiritually impaired (not only for the grave scandal, but also based on the objective heresy they spoke or acted).  But...this would not affect their material office, since obstinacy is necessary for removal from a temporal standpoint.  We all know that +Bellarmine said that we must resist a bad pope and to me, this means we ignore a spiritually impaired pope who speaks objective heresy, even if we don't know if he's obstinate or not (that's for the Church to figure out anyways).  A spiritually impaired pope would be a "doubtful" pope in the sense that he is not to be trusted; his orthodoxy is questionable.  Yet...his material office is still intact, until a material process is followed to remove him.
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Why can we all not agree that material heretic (not yet proven obstinate) pope is to be ignored, resisted, considered as severed (spiritually speaking)?  All of this debating is over what happens to his MATERIAL office, which in the grand scheme of things, is lowly, as compared to his spiritual office.  A non-yet-declared-manifest heretic pope, who still holds the material office, is practically the same as no pope.

Pax, do you see your error here?

What are you talking about?  Go read again the text in red.

Ok, I don't have time to develop this in great detail, so I'll summarize my argument.

S & S hold UPA to be an "infallible sign" that an elected pontiff receiving UPA satisfied all of the conditions of being pope. Well, as Fr. Kramer says, a necessary condition for a pope is possession of the Catholic faith. Thus, S & S are pointing to a post-election occurrence or fact (UPA) as proof of a prior necessary condition.

Well, my position also sees a post-election occurrence or fact as a proof of the lack of a necessary condition, or an "infallible sign" that a condition was not satisfied - namely, the possession of the Catholic faith by the one elected: the post-election occurrence or fact being the promulgation or teaching of error in a "pope's" authoritative magisterium that contradicts Catholic dogma, doctrine or Tradition.

Fr. Boni: “Given the total legal groundlessness of these suppositions, even to want to give credit to the information on which it claims to take root, the bogeyman - rashly agitated - of the current assidarsi on Peter’s chair of a doubtful Pope also vanishes. However, the canonist have constantly and generally chorus that the peaceful "universalis ecclesiae adhaesio" is a sign and infallible effect of a valid election and a legitimate papacy: and the adhesion to Pope Francis of the people of God cannot be put in any way in doubt.” (Dr. Boni, “Beyond a Resignation. The Decision of Pope Benedict XVI and The Law,” Bologna, 2015.)