If he goes there as a kid would he still be Genghis Khan though? But that gets into the whole nature versus nurture debate which is off-topic for the thread.Or you can have Genghis Khan brought as a kid to Constantinople via Trebizond/Samarkand after his captors decide to exile him westwards. I'm looking at you, A Khan in Constantinople...
The Mongols were very adaptable; OTL they built a gigantic navy to defeat the Song Dynasty. If they can defeat perhaps the most powerful and advanced nation on the planet, they can probably defeat the remnants of the Byzantines.
Plus, at this time the Byzantine Empire was ruled by the Catholic Latin Empire. Similar to the "Better the Turban of the Turk than the Tiara of the Pope" ideas which floated about OTL, they may prefer the Mongols to their Catholic overlords and help them depose the Latins.
A Mongol Constantinople, defended by an elite cadre of Bulgar horsemen, sounds like the most badass thing ever.At this point Bulgaria was the vassal of the Golden Horde, and there were several successor states to the Byzantine Empire. Perhaps if the Horde were to offer an alliance with Serbia, they might also gain additional help in conquering Constantinople. Of course, this might also affect the Horde's religious stance when the Mongols would conquer Constantinople. I'd go with the scenario of Sartaq surviving for this to occur.
A Mongol Constantinople, defended by an elite cadre of Bulgar horsemen, sounds like the most badass thing ever.
IF they managed to conquer Constantinople, is there any chance that the Horde will adopt the Orthodox faith?