According to A History of Christendom, vol.2 The Building of Chrisendom by Warren H. Carrol pages 173-178, your timeline and other facts are a little off.
544 - Emperor Justinian issues a decree condemning the Three Chapters and the four Eastern patriarchs, lead by Mennas of Constantinople, give assent to the decree on the condition of the Pope’s approval.
November 2, 545 - Pope Vigilius is arrested for not approving the decree
Fall of 546 - The Pope is brought to Greece when it becomes clear he will not approve the decree
January 547 - The Pope arrives in Constaninople. Lifts his earlier excommunication of Mennas for even conditionally assenting to Justinian’s decree and gives private assurances to Justinian he will issue a document.
April 548 - Issues “Judicatum” condemning the person and writings of Theodore of Mopsuestia, and the heresies in the writings of Theodoret of Cyrrhus and Ibas, without prejudice to Chalcedon. In reaction to this, he is excommunicated by a synod of African bishops.
August 550 - Vigilius convinces Justinian to allow him to withdraw Judicatum and have a council to decide the matter. In exchange, Justinian demands Vigilius’ promise in writing that he will do what he can to convince the council to condemn the Three Chapters.
July 551 - Western bishops state they will not attend the proposed council or if they did attend they would not vote for condemnation. Justinian issues a second statement condemning the Three Chapters and insists the Eastern bishops sign it. Vigilius announces excommunication for any bishop that signs the statement. Many bishops sign and are excommunicated. This is done because it is the Pope and the council that decide issues of orthodoxy and not the Emperor. In reaction the Emperor places Vigilius under closer guard at the Placidia Palace.
December 23, 551 - Vigilius escapes and takes refuge in Chalcedon.
February 5, 552 - Vigilius issues an encyclical denouncing his treatment by Justinian and professing his faith
March 552 - Justinian sends Mennas and Theodore Askidas to Chalcedon to apologize for his mistreatment and pledging fidelity to the Council of Chalcedon and support for the Pope in suppressing the controversy until the council met. Vigilius returns immediately to Constantinople but is not allowed to return to Rome.
May 5, 553 - The Council of Constantinople convenes. Only 166 bishops attend. Because he was unable to hold a preliminary synod in Rome and the small representation of Western bishops, Vigilius refuses to attend but issues a “Constitutum” reaffirming Chalcedon, condemning the works but not the person of Theodore of Mopsuestia, upheld Theodoret of Cyrrhus and Ibas of Edessa, condemned five heretical Nestorian propositions, and directed further discussion of the matter to cease. In response, Justinian issues Vigilius’ written promise to attempt to convince the council to condemn the Three Chapters’. The council takes up the matter.
June 2, 553 - The council anathematizes the Three Chapters but upholds Chalcedon. Theodore of Mopsuestia’s works and person were condemned, only Theodoret of Cyrrhus’ heritical writings are condemned as well as the letter of Ibas of Edessa. They did not decide the issue of whether Ibas of Edessa actually wrote the letter which he apparently denied. Vigilius’ name was struck off the diptychs
June to December, 553 - Vigilius along with his staff are put in close confinement on a diet of bread and water on the island of Proconnesos. During this Vigilius suffered from kidney stones from which he later died on the way back to Rome.
February 554 - Vigilius issues a final Judicatum ratifying the Council of Constantinople, because he said after study and reflection he had come to see how much heresy, whether or not intended, was indeed included in the writings.
None of the heresies were specifically approved by the Council of Chalcedon. The authority of the Pope had been upheld because even the action of the Emperor and a council could not decide the issue without the ratification of the Pope. No fundamental dogma of the Faith was involved in the Three Chapters controversy. Therefore, Vigilius ratified the council.
June 7, 555 - Pope Vigilius dies in agony from kidney stones in Syracuse, Sicily on his way back to Rome after being released from ten years of exile by Emperor Justinian.
Based on that, I see nowhere that the councils or the Pope taught error. Therefore, there is no reason to call the Doctrine of Infallibility into question. After all, the Doctrine of Infallibility does not state the Pope or the Church will never make mistakes only that they will be prevented from teaching error and I see no error that was taught.