well a post 1025 pod means the byzantines still have souther italy (they could of course loose in the late 11th or later centuries when focusing on the east) but i see expansion in to italy as really hard since the papacy hre and others would not like that.
I mean the Papacy was unable to really do much against Norman expansion. Had the Normans been more fortunately they probably could have eaten into papal territory as well, seizing Ancona and maybe bits of the Duchy of Spoletto.
A strong and stable Empire with the Catepanate of Italy bolstered by Sicily would in time be able to re-assert the roman presence in the region.
To prevent the HRE you have to prevent Charlemagne being crowned by the pope and if that happens, Byzantium probably maintains its prestige.
I mean Rhomania still had its prestige. The Carolingian and Ottonian courts basically imitated Roman fashions in Constantinople. Basil II almost arranged a dynastic union with the Holy Roman Emperor. At various points the two Empire cooperated with each other having some sort of mutual understanding with each other.
It was the schism and the sudden fall of Anatolia that really lowered the Empire's prestige in the eyes of the west. Some probably saw the chaos after Manzikert as God punishing the Greeks for their "arrogance/decadence."
Either way what we consider the "modern day" in the world where Rhomania survives and continues its resurgence is basically unrecognizable to us. A good example of this is the dynamics seen in
@Basileus444's tl an Age of Miracles.
The development of philosophies/political ideologies are also quite radically changed. The modern notion of the Westminster style Constitutional Monarchy, stemmed from the old feudal customs where the monarch's power's would be limited by his other nobles. The old Germanic Kings after all were elected into power and were ruling theoretically by their consent.
In contrast in Rhomania no such concept really existed. The Empire essentially was a Republican Monarchy where the Emperor and his power was beholden to the "Senate and people of Rome." In practice and by design the Imperial Senate was little more than a ceremonial institution functioning as a glorified city council. Though in periods of crisis it was able to throw in its weight if the Emperor requested its advice, or say the matte of succession needed to be settled. The Emperor in his power was absolute, though he was expected to "obey the laws of the land," providing just government lest he lose the favor of the Romans and find himself deposed.
The Empire would probably keep its autocratic nature in terms of its government without much difficulty. Though Rome surviving would definitely affect the Renaissance which somewhat romanticized the idea of the Roman Republic. Such notions would likely have cold water splashed in the East as the Empire was still the legal continuation of the "Res Publica." I wouldn't be surprised if Roman historians paint it as largely a period of chaos and instability which inevitably led to the rise of the Empire to most effectively manage Rhomania's government and vast territories. As the separation of Church wasn't really a concept within the Empire, the Church would almost certainly be an important pillar in Imperial society and government. The Empire controlling most of the Patriarchates of the Pentarchy would also have a massive effect on Christendom as well. In such a scenario I don't think the schism would ever really come to pass. More likely the Pope might try to ingratiate himself with the Eastern Emperor to better play him and the Holy Roman Emperor off each other to preserve his position and the Papal States' territorial integrity.
The Roman method of succession was a double edged sword simultaneously providing massive boons and massive setbacks to the Empire. Often times decadent rulers or dynasties were overthrown in favor of better ones like the Komnenoi, Macedonians, Heraclians, etc, but it could also lead to chaos like the crapshow that occurred after Emperor Maurice's deposition or even Emperor Andronikos I's deposition. Andronikos while harsh was doing a hard crackdown on aristocratic corruption/decadence occurring among the Imperial Government. This was largely thanks to the deals Alexios I and his successors made with the aristocracy which in the short term provided the Empire quickly needed funds in a time of crisis. But as the Empire's finances recovered in the early 13th Century, these policies were becoming a liability for the government.
The best way for the Romans to survive into the modern day is simply to have them have a long lasting dynasty allowing for its Emperors to gradually initiate a policy of de-facto and finally de-jure hereditary succession. This almost happened in Roman history, but the Macedonians and Heraclians had a great deal of bad luck at the end which could have easily been avoided. Basil II having a son for example, allows the Empire to pass on to a trained heir rather than Basil's idiotic and hedonistic brother. The Macedonians had a massive following among the Empire's subjects. When news reached the ears of the people that Theodora was going to be assassinated, the people stormed the monastery she was in and despite her protests, clothed her in the Imperial purple and coronated her as co-Empress with her sister Zoe. The mob paraded her down to the Imperial palace and demanded that she be reinstated to the throne.
The French Capetians who started off in a pretty horrible position, over the centuries managed to establish stable hereditary succession, so its not implausible to assume that the Romans wouldn't be able to do the same.