^Another thing-while the idea of having Osman I convert to Christianity and found a family called the Osmanoi is ironically appealing, I'm not sure it works. "Osman" is an Islamic name (derived from the third Caliph, Uthman-Ottoman Turkish, probably due to Persian influence, transformed Arabic short "u" into "o" and Arabic "th" into "s"), so it seems likely to me that the name would be dropped for a biblical name on converstion to Christianity.
Beyond that, the Ottoman dynasty was founded by a man named Ertogrul, who supposedly led a group of Turkic cavalry out of Central Asia (probably fleeing the Mongols), and into Anatolia, where they took service under the Seljuks and were granted some land. Ertogrul supposedly arrived in Anatolia in 1227, so in this TL he might well work for the Byzantines, become a Christian, and give his son (OTL Osman) a Christian, rather than Islamic, name.