I disagree. Behold this image from the Arch of Titus in Rome:
This could very well be a depiction from Titus's triumphal entry into Rome after the sack. Furthermore, Procopius was an eyewitness to the 6th century campaign in Carthage and Belisarius's subsequent triumph in Constantinople and would, I think, be unlikely to make up such a story.
I think this notion that the temple treasures are in the possession of the Vatican stems from a document penned by an anonymous cleric from 1073 which states that the Ark of the Covenant, the menorah, and other treasures are kept in the Lateran Basilica in Rome. Personally, I doubt it.
If I had to guess, I'd say that they were either destroyed by the Muslims during the 7th-10th century or else melted down during the crusader sack of Constantinople in 1204.