For some hand-wave-y reason, the Turks of Rum convert to Orthodox Christianity during the 12th century. Then, a few hundred years later, some Turkish nobleman founds an Ottoman Empire analogue, which goes on to take Constantinople from the Byzantines.
Now, IOTL Mehmet the Conqueror and his successors occasionally used the title Kaisar i-Rum (Caesar of Rome), but it was never a particularly important title and neither contemporaries nor later historians really considered the Ottoman Empire to be a continuation of the Roman/Byzantine Empire. Basically, I was wondering whether this would still be the case if the *Ottomans were Christian: would their rulers play up the "heirs of Rome" angle more, and would other people, both contemporary and in later times, view their state as the genuine and legitimate continuation of the Roman Empire?
Now, IOTL Mehmet the Conqueror and his successors occasionally used the title Kaisar i-Rum (Caesar of Rome), but it was never a particularly important title and neither contemporaries nor later historians really considered the Ottoman Empire to be a continuation of the Roman/Byzantine Empire. Basically, I was wondering whether this would still be the case if the *Ottomans were Christian: would their rulers play up the "heirs of Rome" angle more, and would other people, both contemporary and in later times, view their state as the genuine and legitimate continuation of the Roman Empire?