There were several sacks of Rome prior to its peaceful cession to Odoacer in 476. The bishop of Rome had to plead for retaining jurisdiction of Vatican Hill for the next centuries with whichever Arian or Orthodox took political control of it. While he still probably represented a hypothetical continuous elite, he had to remain in full communion with the rest of the Pentarchy and could only assert his privilege over them by allegation. It used to be that other bishops could contradict the pope successfully and talk him down, but over the centuries he kept asserting that they couldn't until he finally had them excommunicated in 1054 via his legate, thus losing four of the five Pentarchy sees.

What really got the pope going into his own power was getting land grants (the Papal States), via seizing his own political control of duchies. This began in 756 after years of Pope Zachary supporting Pepin the Younger and getting the Duchy of Rome in return. This emboldened Leo to announce the HRE on Christmas 800. I'm not sure what region of the west you're thinking of that was controlled by the east between 480 and 756, though there probably was a holding here or there in that time that was nominally Byzantine.

Turks (Seljuks) didn't come to Turkey from Central Asia until the 11th century (founded as Ottomans in the 13th). Those first called Hungarians and Austrians were basically just margraves among the early HRE. So it's a long road from the sacks of the 4th to 6th centuries to these later aspiring conquerors. As I said, all that I've seen was that whenever the pope's locals had any influence it was either to stay in Rome under its present power or to bring another power to take over Rome. I don't see a narrative for migration of power from Rome elsewhere, only of Rome picking winners among whomever was asserting themselves at the moment. I do appreciate the opportunity for brushing up and new facts.