Ladislaus: No, they cannot depose him. Only God deposes popes. Bellarmine is very clear about that. It's only because the form of the pontificate has already left him that can exercise any authority over the ex-pope.
Roman Theo: You are partly correct and partly wrong. It is true that God alone deposes a pope authoritatively. The Church “deposes” him legally by declaring him deposed. Prior to a declaration of the Church, he must be obeyed in all legitimate commands, as Bellarmine teaches in the quotation I cited above.
You are also correct to say the Church cannot exercise any authority over the pope as he retains the form (jurisdiction) of the pontificate, but the form is not removed without human judgment. Bellarmine addresses this point directly:
“For Jurisdiction is certainly given to the Pontiff by God, but with the agreement of men, as is obvious; because this man, who beforehand was not Pope, has from men that he would begin to be Pope, therefore, he is not removed by God unless it is through men. But a secret heretic cannot be judged by men…”
Papal jurisdiction (the form) is taken from the pope by God, but not without the judgment of men.
Ladislaus: With regard to this imperfect judgment, the discernment of fact with regard to heresy becoming "manifest", consider it from the perspective of the following scenario:
On May 1, 2018, Bergoglio declares, "I know that the existence of hell is Church dogma, but I simply don't believe it." [not entirely implausible given recent events]
On May 15, 2018, Bergoglio defines a dogma.
On May 31, 2018, an Imperfect Council convenes and makes a declaration: "On May 1, 2018, Jorge Bergoglio ceased to be a member of the Catholic Church and therefore to be Pope." You know, it would take a while to assemble everyone.
So did Bergoglio lose the office on May 1 or on May 31? May 1 obviously.
Roman Theo: No, that is not obvious. St. Alphonsus says a pope must persist in notorious heresy to lose his office. One formally heretical statement does not qualify as persistent notorious heresy. In the hypothesis you presented we fall back on the teaching of Bellarmine, that a pope must be obeyed in all legitimate commands until he is legitimately declared to no longer be pope. So, in accord with Bellarmine’s teaching, as far as we are concerned the pope would cease to be pope on May 31st, not May 1st. And the dogma he defined on May 15th would be protected by the infallibility of the office he legally holds.
Ladislaus: In other words, the discernment of fact FOLLOWS upon the actual deposition by God and not the other way around.
Roman Theo: But if he had already lost his office, why would the judgment be limited to a discretionary judgment, rather than a perfect judgment?
The discretionary judgment comes first. Once the fact of his heresy has been sufficiently established and he is deemed to have lost his office, then the Church is permitted to judge and punish him, which is proper to a perfect judgment.