It sounds a little far fetched that the Christian church would begrudge the Roman Empire for their persecution, in light of the fact that there had been Christian emperors for more than a hundred years prior to the Western Empire's demise. Where do you get this from?
From accounts of the church in Rome up to this period. From the writings of the Church fathers, their organization, construction and pilgrimage patterns in the city, etc. From their responses to existing Byzantine emperors when the latter tried to flatter the pope with the connection.
Half those 'Christian' emperors were heretics and schismatics, and persecutors in their own right. With a couple of exceptions, they weren't really held in good memory.
Roman Church was obsessed with martyrology. That was its organizing 'character'. Just check out the Christian calendar - every day celebrates the Roman persecutions. Persecution was their organizing ethos. There was the Rome of Peter and Paul, to be cherished, remembered, while the Rome of Augustus and Nero was to be forgotten and disdained. The Church wanted nothing, absolutely nothing, to do with imperial Rome.
What would a Christian visitor do in Rome? He would visit the titular homes of the Christian martyrs - Cecilia, Lucina, Vestina, Prassede, Pudenzia, Prisca, Bizante, Pammachius, Pastore, Clemente, Nicomede, Anastasia, Balbina, Nereus and Achilleus, the brothers John and Paul, priests Marcellinus and Peter, Chrysogonus, Cyriacus, the Quattro Coronati, the popes Clement, Sixtus, etc., whose names and stories they knew by heart from the church calendar and services. They visited the shrines of the Apostles St. Peter and St. Paul, St. Lawrence the Archdeacon, St. Agnes the Virgin, the Mammertine prison, the catacombs of Trastevere, etc. And at every step the story of the Roman persecutions would be told and retold, again and again.
And while they went from martyr to martyr, the hills of the Palatine and Capitol, with their pagan temples and palaces of the persecutors, loomed in the background, avoided, disdained.
And its not as if those imperial institutions disappeared either. Up until the 7th C., the senate, the forum and the imperial palaces on the Capitol-Palatine area remained in operation - albeit in the hands of a new oppressive enemy, the Arians (whether Byzzie or Ostrogoth). The hill's pagan temples continued to be focal points for crypto-pagan worship down to the 6th C. And, of course, that most despised of monuments, the Colloseum, the site of so many Christian deaths, continued to be operated by the imperial authorities until the 6th C.
What would the Church, what would a good Christian, want to do with
that?
Now, admittedly there was a change under Gregory the Great, who absconded with administrative authority in the city, and did try to claim a bit of Roman authority for himself. But he orchestrated this very carefully, to avoid poking at the Church's raw sensitivities. He did not claim imperial title directly, but rather installed the seven deaconates in or around the abandoned pagan hills (seven deacons was the civic organization of Apostolic Jerusalem and the conscious reference point for organization of Christian communities). By this greatly symbolic gesture, Gregory asserted the final 'victory' of Apostolic Jerusalem over Imperial Rome, giving him the 'right' to claim the civic authority of the latter without violating the living memory of the martyrs.
It was only after this that the Church felt comfortable enough to begin toning down its inherent hatred of the empire and everything related to it and bridging the hitherto inimical chasm between Roman Church and Roman State. But the martyrology continued to be drummed, and the connections remained touch-and-go.
(On the Pantheon: the emperors had issued edicts at least as early as 408 allowing the conversion of pagan temples to Christian use. The Church refused to make use of it. Not a single pagan temple was converted, until after Gregory - the Pantheon in 609 was the first.)
(And, alas, we can also credit - or rather blame - Gregory for pushing that lamentable rewrite of history asserting it was the Jews, not the Romans, who were the Christ-killers. Part of his whitewash of Roman heritage.)
Now, it is true this changed even more dramatically in the late 8th C., which is when the whole Donation of Constantine thing was cooked up, and the pope and Roman cardinalate claimed direct Roman authority openly as 'successors' of the Roman Senate and Republic.
But emperors qua emperors still weren't that cool - still too much Nero, as it were. Charlemagne's measure, whom he was to be compared to, was narrowly Constantine, the only 'good emperor' in Christian estimation. The allusion was not to the 'glory' of pagan imperial Rome, but to the magnificence of Constantine alone.
