Crusader Kingdom?
With as much as it is often bandied between Turks, Mamelukes, and Persians, the poor province of Judea could certainly be put to better use as a Christian kingdom, the remnants perhaps of the Crusaders of old ... the Kingdom of Jerusalem, never taken by the heathens in their malevolent conquest of the Holy Land? I think that would afford a perfect staging point for the Catholic powers into the very heart of pagan Asia Minor, and probably make for some interesting monarchs and leaders as well, if the colourful politics which were often seen in the Crusader kingdoms are any indication. I realize we already have the Byzantium option, and this is more a bit of whimsy and the avid Medieval historian in me than anything else, but if it sounds plausible, I'd be happy to assist in the research and suggestion of historical data.