The Seljuk Turks took Anatolia after the Battle of Manzikert and it was partially recovered by the Komnenoi. This recovery stagnated after the Komnenoi were deposed. Let's say a new emperor comes along in the 1200's and retakes Anatolia. In this POD the Venetians fail in taking Constantinople and the Byzantines are very angry and rally behind a new leader who deposes the Angeloi. How would the re-integration work? How much of Anatolia was Turkish? When did Anatolia start losing its Greek and Orthodox identity?
I think it'd be Okay, actually, at the date you have chosen. Most of the region was still Greek. The Turkish presence was mainly in the region near Ankara, Iconium, and the Halys river, in central Anatolia. The north and south coasts and the western third of Anatolia were still Byzantine. Central Anatolian plateau was Turkish, although some Greeks probably still existed. For instance at Lake Pousgous we know there were Greeks in the 1130s, although they preferred to fight for the Seljuks against the emperor John II Komnenos (interestingly!). Eastern Anatolia would have been a complete mix of Greeks, Turks and Armenians, under Turkish rule.
As I understand it, Anatolia as a whole only really shifted to becoming majority Turkish in the 14th century or later. But a more specific answer does depend on which area exactly, as Anatolia is a big place. Some areas remained majority Greek until as late as the 1920s.
Also, one interesting detail I learned about the emperor Manuel I Komnenos (1143-1180). In his time, the Seljuk court and the Byzantine society saw a lot of mixing between Turks and Greeks, and they were found in both territories. Some Greek nobles became Seljuks, and some Turks joined the imperial army and even joined the imperial household and became 'Greek'. In this context, the emperor attempted to introduce a reform to the conversion process from Islam to Christianity, to make it easier for Turks to convert to Christianity. Specifically, he wanted to remove the part where they were required to "reject" the god that they believed in before.
Now, from a theological perspective, as I understand it the three Abrahamic religions Islam, Christianity and Judaism already worship the same god, so the process of "rejecting" the god of Islam doesn't really make much sense any way, since it is the same god the Christians believe in. But anyway, Manuel's proposal would have made it easier to convert, because Muslims wouldn't be required to say they "reject" god they believe in, instead they'd just accept that Christianity is the "true" way to reach that god and accept that they were misguided before. Which is a lot easier than requiring someone to actually "reject" the god they believe in entirely.
So as you might have guessed, I think Manuel's idea was great and would have been a very positive step for Byzantium. Unfortunately however, I understand that hardline Orthodox clergy rejected the proposal, and it was never implemented. In my personal opinion, these narrow-minded clerics and their prejudice were an obstacle that should have been overruled. Had the change been introduced, and combined with more successful Byzantine campaigns in the Anatolian plateau, one can well imagine that the capture of Konya (for instance) might well have resulted in the restoration of Byzantine rule over the interior, followed by absorption of the Turkish population and assimilation. After a few generations, it would be as though Manzikert never happened.
Edit - also
@CalBear could we get this thread moved to the pre-1900 section? I believe the OP posted a thread where he asked how he could do this. Thanks!