From some quick searching, it sounds like the Kurds had been there basically forever, in some form or another. Their generally accepted primogenitors were present since the Bronze Age.
The Kurds might be a used by the Byzantines as a buffer against the troublesome Armenians and any resurgent Iranian nation.
I think the best way to have the Byzantine Empire only include the modern day Turkish boundaries is to have some type of left/right civil war between 1880 and 1950. If the Byzantine Empire sticks to the path of the Russian Empire they might be an revolution likely a left wing one.
If the Emperor can maintain support of some of military (perhaps by some program or another like the Theme System) they will be able to keep order in Asia and hold Constantinople firmly.
If the Empire suffers some type of Naval Disaster in the Aegean at the hand of the European Greeks that might buy the revolutionaries some time however this is going to be a war with two sets of alliances- the status of the outside world will decide who fights for whom.
If both sides fight sufficiently to a stale mate after two or three years a fragile compromise between the Empire and the Republic might osensue what factors that could keep this compromise alive are up to you.. Nuclear tensions are the easiest way to do this but not the only way..