We also need to realize that it was during this period that the Carolingian Empire (forerunner of the Holy Roman Empire, Kingdom of France and other minor French and Italian kingdoms) was solidifying control over much of western Europe. Keep in mind that Charlemagne certainly recognized papal primacy when he first protected the Pope and later was crowne Emperor, he certainly wasn't operating under the belief that the Byzantine Emperor was in charge.
Now, the Carolingian emperors and the Byzantine emperors were certianly rivals, but there was never any direct conflict between them. However, there were definitely some tense periods.
Nevertheless, the fact remains that we are being asked to believe that the Byzantine emperors controlled the Church and VOLUNTARILY handed that control over to the Papacy (if it wasn't voluntary, we need to be told what specific event precipitated it). Now, why would a Byzantine Emperor give control to a Pope who was over A THOUSAND MILES AWAY? A Pope who was far more likely to take the side of western European rulers in any dispute. If the theory being offered was that the Byzantine Emperor ceded control to the Patriarch of Constantinople it MIGHT be believable, but not the Pope.
Charlemagne wasn’t operating under the belief that the Pope was in charge either.