Charlemagne was playing with fire here -- the Byzantines were the big kids on the block at this point, not him. Due to the lack of a Frankish navy Charlemagne famously failed to conquer even Venice, right off the Italian coast, despite its considerable distance from the rest of Byzantium's territories. Venice would remain independent from the western empire for the rest of its history and its various incarnations. Though the Franks (and later the Saxons) clashed with the Byzantines a number of times over southern Italy (where the Byzantines still had possessions after the reconquests of Justinian in the sixth century), to my knowledge no Frankish emperor ever took the fight east, or attempted to conquer a "reunified" Empire in any way. Charlemagne and his successors were happy with a begrudging acceptance of their claim to empire from their contemporaries in Constantinople; the Saxon emperor Otto the Great went so far as to negotiate a very hard-won marriage for his son Otto II with the princess Theophanu in the 960s in part to get a Byzantine rubber-stamp on his rule (this would later result in Theophanu as queen-regent for Otto III signing documents as "Emperor Theophanu," which is some cool gender shit if there ever was some)

I can't speak to the political leanings of the pope involved, not least because it's not my speciality. I can say that Irene was a crazy person and probably many even in the Byzantine court considered her illegitimate; she was one of the more ruthless rulers of Constantinople. Nevertheless, Charlemagne never to my knowledge harbored ambitions for the reunification of the Roman Empire; by his time the two halves were well and truly distinct and I'm not sure he really would have known what to do with Greece even if he had conquered it.

Hope that's thorough enough -- will obviously answer further questions if asked

edited to remove self as source. I don't have a specific source from this one because it's not really a question that's asked specifically in the field (bit of a contrapositive). For further reading I could suggest Tim Reuter, Germany in the Early Middle Ages c. 800-1056 (Longman, 1991)

The furthest east the Franks came was roughly the area of a roman province of Dalmatia, just about the time of late Charlemagne. There they started to threaten Bizantinian coastal city hubs, which were important to the Byzantines, for retaining the connection with most of eastern and northern Italy. The tensions rose untill the peace treaty of Aachen (812). There they reached an agreement where Byzantium keeps the coast, and the inland areas (populated with Croats since early 7th century) belong to Frankish sphere of influence. Soon after Charlemagne died, Frankish power began to wane and by the mid 9th century, Croatia was a sovereign duchy (under dux Trpimir), ruling over inland areas that were untill recently under Frankish rule. anouther 70 years and Croatia controlls the coast too and all possible direct contact between Franks and Byzantines stops.
So my answer would be no, he was not in a position to possibly threaten Byzantines in any serious way. Any offensive on his side would probably face an even harder retribution. Even though Byzantium was already on a downward power-trajectory, it was still a major European/Mediterranean power.

Lit: Dzino, Danijel (2010) - Becoming Slav, Becoming Croat: Identity Transformations in Post-Roman and Early Medieval Dalmatia
Ante Milošević (2000) - Hrvati i Karolinzi

Can I ask a follow up question to this? I think rules say it is possible, while there is also an automoderator deleting short comments. Anyway, I'll try:

Pope did not consider Empress Irene of Constantinople to be legitimate, and thus the imperial throne was vacant at the time.

How come Pope's consideration viewed as decisive for imperial throne and when does it start? I think(but I don't know) Byzantines were rightful successors of Roman Emperors, Western parts were lost to the 'barbarians', and at some time Pope legitimazed them to gain power himself. (Although I admit this is my weak interpretation). Can someone explain me the real situation?