Even if Charlemagne could somehow conquer Constantinople, or gain somehow power in Constantinople
(for example by a marriage with Empress Irene), it would be really difficult to manage the two empires.
(I think it is much more plausible to arrange a marriage between him and Irene (but still not likely) than conquer Constantinople, for conquering Constantinople, you need a large fleet and a large army against the strong walls of the city.)
If Charlemagne gain the control over Byzantium, some really interesting problems could be the result. His new empire would be far to big, it was even difficult (especially for his successors) to manage the OTL Frankish Empire.
It is impossible to control the Byzantine Empire from the core regions of the Frankish Empire (places like Aachen). Traveling between Constantinople and Aachen would be also very difficult.
If Charlemagne decided to reign in Rome or Constantinople, he maybe could reign over the Byzantine Empire, but he is a foreigner and he don't knew the aristocracy of Byzantium, therefore his reign is weak, and he needed constant attention to hold control over them.
At the same time he would loose much control over his Frankish Empire. Without a powerful Emperor, the Saxons probably could defend their independence. At the same time powerful local families could gain control over large regions and fight against each other.
If Charlemagne decided to stay in Constantinople, or reign from Italy, someone should govern the Frankish core regions (Westfrancia), maybe he could create here a kingdom for his son or some relatives.
The result would be a short reign of Charlemagne over Constantinople (but he or his successor would be propably replaced by a local family) and a weakened Frankish empire.
The pope was weak in the early middle ages. Under the right circumstances, he could support Charlemagne against Byzantium. Maybe that would be nice for Charlemagne, and could strengthened his position in Italy, but it won't decide the war.
An earlier schism seems unlikely to me, and such a schism would weakened the position of Charlemagne in Constantinople.
But a conflict between Western Christianity and Iconoclasm would be possible. (It would be also possible that some Iconodules could support Charlemagne against the Iconoclasts)
And if such a schism happens, it would even
(In OTL the papacy turned from the Greeks to the Francs in the 8. century, because only the Francs could protect the Papacy against the Langobards.)
I don't know much about the Abbasids, but I would be first frightened if such a massive united empire forms at my border. But after some time, they maybe realized that this empire is to large and weak. They could use internal unrest for their goals.