in theory, one could play as Byzantium in EU2 and then convert it over on the ol' Eu2Vic utility. no need for tortured imagined histories, as one will have a plausible history dating back to at least 1419 (or 1066 if you've CK) to work with. it's not perfect, but it is simple. instead of simply using greece to do the job, i created a new user define Byzantium, and created a suitably edited version of the ottoman political parties as well.
if you really wanted events, well, i guess the pressures on a 19th century byzantium would be much the same as those that faced the ottomans (and other unweidly, multinational empires like austria and russia), so many of the events can reasonably be transposed, with alterations.
the axis of the empire would be westward toward europe rather than eastward toward the islamic hinterland, i guess, but i suspect that, like russia, the state symbol of the double headed eagle would represent the political and social dichotomy of byzantium, i.e. are they of the east or the west?
possible cleavage patterns:
secular/religious
military/civil
islamic/christian
modernist/traditional
much the same as the ottomans or russia, indeed. which is why i transposed the ottoman parties, with suitable alterations. the people involved would largely be the same, after all.
the russians were never particularly shy about conquering and partitioning and subjugating their fellow slavic imperialists (poland, ukraine, etc.), so there is little reason why having a fellow orthodox emperor rather than a muslim sultan on the bosporus would have stopped their ambition to obtain a port on the med. they would simply have found an alternative pretext, i guess.
i guess that by the late 19th century, the "emperor of the romans" would come to resemble all of the other monarchs of the period: an unhealthy, consumptive dimwit more easily amused by shooting, shagging and eating things, who was largely of german extraction, and probably related to Queen Victoria. i guess if revolution came, it would take the guise of the "second republic" or something.
anyway, here's one i made earlier, to coin a phrase.