Not so much a question about how the empire could survive, but a question about if it had. Since I don't want to suffer from the question of how it survives, I'll briefly detail a scenario that fulfills my requirements.
We have a successful Alexios II (say Manuel I lives an extra decade and nothing really bad happens in that time), thus avoiding both the Fourth Crusade and the loss of Bulgaria. Now, despite the weakening of the Seljuks, the empire doesn't reconquer all of central Anatolia, the population is more thuroughly Muslim and Turkish, and between the Italians and the Mongols and the unrest in Serbia and Bulgaria the empire just never manages to retake these areas, and ultimately becomes content with using them as a buffer state against the powerful Persia based dynasties that tended to crop up intermitently in the medieval and early modern near east. Without the sack of Constantinople, relations with the west improve, either the schism is healed, or its importance in the public mind wains over time. Byzantine borders remain fairly constant, perhaps a little bit of loss along the southern coast of Anatolia and the Adriatic, but overall it retains modern Greece, European Turkey, Bulgaria, most of Serbia, Albania, Macedonia, a toehold in the southern Crimea, the Western half of Anatolia, and eastern Anatolia north of the Pontic Mountains. These regions have all come to be a fairly unified nation, with regional languages having been cut down in number of speakers (though larger ones are still extant) and orthodox being around 85% of the overall population.
We'll assume the industrial revolution comes around the same time as OTL, and that the Americas have been discovered and colonized, basically that things are quite close to OTL overall with the exception of surviving Byzantium.
With this situation, how powerful can Byzantium be in the industrial era (say 1850 onwards for simplicity)? Thinking in terms of great powers from the OTL latter 19th century (Britain, France, Germany, Austria-Hungary, Ect.), who would they be most like? Would the Balkans' population be appreciably greater with added stability and no Ottomans? Lastly, what would the empire's goals be? Do they attempt to create a Middle Eastern/North African colonial empire, or look further afield to colonize, or just focus on Europe? Would they attempt to take control of an Alt-Seuz Canal and use that as a gateway to further colonization?
Basicall, in a nutshell, what would this Byzantium be like from 1850 to the present?