The premise of this question: The Komnenoi restoration manages to more or less reestablish the borders of the empire as they were at the end of Basil II's reign by the mid 13th century. Naturally some areas are more secure than others, and this does not mean that the interior of Anatolia isn't Turkified to a strong extent.
So would it likely seek to take and/or control the same areas (more or less) as the Ottomans did OTL? Not necessarily right away, but would that be a goal for an ambitious emperor? Would it be more concerned with the East? (the way the Western European powers were in the 16th century and beyond)? How would the Latin powers of Europe see it (as opposed to how the Ottomans got a very bad name as the Terrible Turks)?
Basically, how would it be like (and unlike) the Ottoman state regarding strengths, weaknesses, enemies and ambitions.
It seems that it would probably be different in more ways than just having the Hagia Sophia as a church, but beyond that...well, hopefully there are resources out there worth reading to get an idea of how this could have looked. Unfortunately, just reading a history of the Byzantines doesn't give a very clear picture of what could have happened had things not gone wrong from 1180 (or earlier) on.