Granted a lot of my knowledge of Ῥωμανία is from it's earlier portion (636-1081 or so) but I think for a model of the Roman state continuing into the modern era we need to look at the Russians. The Romans would suffer much of the same problems that Russia faced when trying to get "into Europe" so to speak. They would be viewed as "oriental" and "eastern" in the eyes of the West and would have a difficult time, at least before the 18th century, in participating in wider European politics. The dynamic would be to look at them as foreigners and schismatics. I think that much of the enmity between the Romans and the "Franks" would still continue at least past 1453 into the 16th and 17th centuries. Much like the Ottomans, perhaps powers like France would use them as a counterbalance against the Holy Roman Empire and, depending how much the Romans expand, they could play a big role in Italian politics once again.
As shown by the Byzantine recovery in the 9th and 10th centuries on to the apogee under Basileos II, the manner in which the Romans deal with the Turks and Timur and how they expand will play an important role in how the state develops. I think as the state expands, the pull away from Constantinople will cause resentment much like it did in the 10th century. Cosmopolitan Romans of the capital will contest power with the land-owning provincials. Perhaps this trend would wane as technology got better and communication and travel allowed a closer cooperation and so a larger range of Constantinoplian projection, but maybe not.
If you want a more developed answer you might need to provide additional details to the timeline and how it diverges but the premise here is interesting. The Republican institutions in the "Πολιτεία τῶν Ῥωμαίων" and their reaction to the passage of time and the modern world are also fascinating. The rise of the printing press I think would involve provincials far more in the horse-tradings inside the capital. Instead of the cosmopolitans often making or breaking a ruler, the provincials would receive news and ideas faster and easier. Constantinople could no longer be an inside political club commanding the provinces. Instead, the significance of public opinion in the themes would matter more since more citizens would be in the loop as to the political goings on and far faster.
The powers of the Basileus would probably stay continuous as is the tradition, the bureaucracy likely get bigger with way more people to take care of and more mouths to feed.
The relations with the East would be more hard to predict. The situation would largely depend on which powers exist there as always but if push comes to shove, the Taurus is likely the consistent political border upon which any further major incursions would fail.
I think the comparison to the Russians is good, but insufficient. The Romans are not a people of megalomaniac nobles ruling simpering Slavic serfs. The peasantry is free (thanks be to God and Leo III), amd the rising urban peoples are enough counterbalance to the aristocrats and beuracrats to keep the power struggles violent enough to prevent a plutocracy. Remember that the Roman government is very socialistic/democratic (before the Komemnoi) and thus the nation will continue to be a very politically expressive (and violent) land.
I agree with your statement about the east. The Romans will pick their battles with infidels ratger than heretics. Even without a fourth crusade, they will grow to see the Latins as demons just as the Russians did. Idk about a westernisation a la Peter the Great, but aside from the political, economic, and religious outlook, Greeks did trade extensively with the West, so i imagine they will continue to do so. They at least will not lag behind the West in terms of molitary, as the Romans (before the Latin occupation, and even after, as they did use canons) were astute observers of what made for sound military strategy and technology. I'm planning on having a back and forth with the Muslim east and the Romans, particularly with the Egyptians, so as to keep the Romans on their toes and from sliding into comfortable domination. Thus it keeps the people from sliding into decadance as can be seen in America.
They will be the most odious of statist societies, making this hard core libertarian cringe. They will depsise any and all of the political theory of the West. They will laugh at their democracies, pointing out the plutocratic nature of them.
As for the West... with feudalism already quite established, the trajectory is set for the plutocratic governments and capitalistic societies that will come. The hierachies are in place, one need only to introduce those historical inventions for the historical outcomes to be the same, the European aristocrats to become entrenched. Add in firearms and centralised states will emerge. These states tax and spend until an Enlightenment happens, to counteract and critique the centralised states. Revolutions will follow. Reactionaries will react to said revolutions and will win or not. Greed is ever prevalent in man, so I am certian colonialism will follow. More centralisation occurs, and an industrial revolution will spark from the progress of technology amd the resources made available to the Europeans. Epidemics will pave the way for more modern science, as will escalted conflicts. Eventually this will culminate in some catastrophic war that shifts rhe balance of power worldwide. My thoughts on history.
