If the Emperor himself is not a Turk or Arab, and his trappings are very European, he could be seen as Emperor of Rome. Just a Muslim Emperor.
This argument itself comes from Eurocentric view/culture. I'm not accusing you of anything of course! Just the grounds of the argument.
In OTL itself there were no Byzantine emperors with what we'd call mostly European trappings. The administration was very centralized and featured governors, major landowners, generals, eunuchs, patriarchs, chariot racers, caulkers, and even many women heavily involved in intrigue and affairs of state, as opposed to the overwhelming landed aristocracy of western Europe. Then there are eunuchs, abject abomination to Romans and early Christians alike if I'm not mistaken, then borrowed by the Christian Roman empire from her Eastern enemies.
Then there were emperors from Isauria, basically savages to the Byzantines, and from Syria, and from Armenia. And some heretics to boot, especially the more mystical iconoclasts.
The Byzantine Empire wasn't just another angle of the Roman Empire, it was a totally different animal and does not fit well into what we call East or West. But an emperor with your idea of very European trappings would have to be a Frank.