I see no reason why a surviving Byzantine Empire, especially one that does not lose many of its richest territories to Arab invasions, cannot be at minimum as successful as the Ottomans in terms of territorial expansion and stability. In fact, I'd argue that the Byzantines are better poised to do so compared to their Ottoman counterparts, in large part due to their assimilationist tendencies and geopolitical advantages. Our hypothetical Byzantines don't have to deal with many of the geopolitical challenges that plagued the Ottomans in OTL, such as revulsion by the Western powers and a big angry Russia breathing down their neck. Assuming the Great Schism is prevented and the Rus adopt Orthodoxy as in OTL, it's possible for the Eastern Romans to be on cordial terms with their Christian neighbors instead of being viewed as an existential threat as the Turks were. With these geopolitical threats removed and with greater access to Western technology and military developments, the Byzantines are well-positioned to be the long-term undisputed masters of the Eastern Mediterranean and the Middle East, with firm control over the Balkans, Anatolia, Levant, Egypt, Mesopotamia, and the Caucuses -- all united under a common Roman civic identity. I suppose there's an argument to be made that the age of nationalism would cause the state to splinter, but I just don't see it. Nationalism didn't really start taking off until after Napoleon, and if the Byzantines haven't made good Romans out of their subjects in the almost 1300 years between the Arab invasions and 1900, then they will have seriously dropped the ball.
This isn't to say that everything will be smooth-sailing for the Romans, or that every region they conquer will happily cast aside their identity for a shiny new one, but it will get done. With enough oppression, state brutality, ethnic cleansing, and over a thousand years to work with, the Romans absolutely can maintain at the very least the Eastern Mediterranean indefinitely.