One of the defining features of Charlemagne's reign was that he was recognised as Augustus, which pissed off the existing Roman Empire based in Constantinople. So I was wondering how, and what the consequences would be, if Charlemagne agreed only to be crowned as Augustus of the West, and recognised by the Romans of Constantinople as such?
There was no "Augustus of the West." The Roman Empire was one, and it had been one for centuries. Neither Charlemagne, nor the Pope, nor the Byzantine Emperors had a notion of Roman
imperium as being something other than a single entity, and its holder as the sole pinnacle of earthly power. Even if Charles and the Pope were to revise their notion of
imperium and declare that the diarchy was back in town and Charles was "just" the Roman Emperor ruling in the west, there's no reason why the Byzantines would accept this, and we know this because they didn't. They have no reason to accept that they are now co-equals with barbarians and the Franks have no power to make them. The worst the Franks could ever manage was to mess them up in southern Italy a bit, and even Charlemagne's power extended only weakly and fitfully over the southern half of the peninsula.
In fact the emperors in Constantinople made peace with the idea of the Franks being "emperors" fairly swiftly; they admitted, albeit begrudgingly, Charles and his successors the title of
basileos. They did not, however, accept them as emperors
of the Romans - the Pope's appointee might be a
basileos but he was at best
Basileos Phrangias, not
Basileos Rhomaion. In other words, they were willing to concede the title but would not accept a division of "the empire" - there was but one Rome with one Roman Emperor, and he ruled in Constantinople.
