Are you predicting the Khantate of the Greeks / Romans , a sucessor Orthodox Mongol Khantate that , in this ATL Claim to be the Roman Empire continued and the heir to the Byzantines ? If events turn out well for the Greek Anatolians , this might end up as a Byzantine Wank.
Yep, that's pretty much it.
And even though Anatolia may not end up being a separate khanate (it's not unlikely that it would become a province of the Il-Khanate, or an ATL equivalent thereof), it would still be an important part of wichever Khanate controls it.
What's more, at least in the OTL Il-Khanate, Mongol clans that governed large cities and important parts of the khanate had a tendency to become powerful factions within the khanate, and as the Il-Khanate crumbled, these factions became effectively independant, and some of them became empires in their own right. The Jalayirids are a good example of this.
And in this scenario, Anatolia certainly is wealthy, populous and both politically as well as economically quite important. Both before as well as after Mongol conquest, Byzantine Anatolia is most propably wealthier and and more populous than Georgia and all of the Armenian territories combined, so it is safe to assume that the clan that gains control of Anatolia would become quite powerful within the Khanate.
This is boundt to have interesting consequences, even if the Il-Khanate or its ATL equivalent remain intact. A Christian Anatolia that is ruled by Christianized Mongols would play an important role in the politics of whichever khanate it belongs to, and as a part of the Il-Khanate, it would greatly strenghten the (already relatively strong) political position of Christianity within the Il-Khanate.
Perhaps this would eventually result in the Muslim factions within the Il-Khanate being balanced out by the khanate's Christian factions...
And if the Il-Khanate falls apart ITTL like it did in OTL, with the result that Persia and southern Mesopotamia become dominated by Muslim polities, it is very likely that this Anatolian polity would become independant, and it would be quite possible that the Armenian and Georgian territories, along with (Nestorian) Christian Mongol clans in the western territories, would gravitate towards this Christian Anatolian polity.
That scenario offers some very interesting possebilities, and like Advernt already said, possibly even a (somewhat unusual) Byzantiwank...
Its an interesting scenario, but on the other hand we have the opposite example of the Seljuk Turks conquering Anatolia and the Turkification of Asia Minor which happened in OTL, so...
The main reason why the Seljuks weren't assimilated by the Byzantines in OTL was that the Seljuks were already Muslim when they invaded Anatolia, and converting to the local religion is usually a crucial step in the process of assimilation. In OTL, Islam prevented the Seljuks from converting.
Islam is a religion that's pretty much designed for preventing nomadic conquerors from converting to the religion of their subjects and assimilating.
And when the Mongols invaded the Middle East, the vast majority of them were still pagan, and therefore far more likely to convert to any of the local religions - which they in fact did in OTL Persia, Central Asia, and China, so there's no reason to assume that they wouldn't do so in Anatolia.