I may be wrong but I believe that Greek nationalism up until the 18th and early 19th century was largely "Byzantine" - they ultimately realized that they could better appeal to the West via Classicism (and of course this was a high point of neo-classicism in general) as opposed to relying on Mediaeval claims to heritage and legitimacy.
I believe Tsar Ferdinand of Bulgaria kept a full set of Byzantine regalia with him during the Balkan Wars in case he ever won Constantinople for instance.
Of course this would be merely with a Greater Bulgaria or Greece headquartered in Constantinople, perhaps with some more reactionary ceremonial and regalia.
It depends how you define 'Byzantine'.
The people never claimed that they were Byzantines. They claimed they were Romans. And they were in fact reamnants of the Roman Empire which migrated part of its dominion to the former town of Byzantium (Constantinople later and currently Istanbul in Turkey)
The crossing and merging of Greek and Roman identity occurred when the Latin Empire sacked the Roman Empire in 1204. People then felt abandoned by the West and the Greek-ness prevailed over the Roman-ness. So the Roman Empire slowly became introduced to the easterners as 'Kingdom of the Greeks' and the term 'Rum' (Roman) became the same as Greek and Ionian(Yunan).
It is true that Greek nationalism at the time was a bit confusing. First, there were those who were Hellenic, ie not in favor of a Christian state and then there were those who were claiming Christianity and Rome, ie the Byzantium region as the home of the resurrected empire. The first also held claim on the Byzantine region (eastern Thrace as we called it) as well as Ionia (Asia Minor, basically the region the Turks know call 'Aegean').
At that time, some of the people behind the Revolution, mostly Greeks, were in favor of a Balkan-dominant state without the aim of a nation-state but an anti-Ottoman empire that would become a leader in the region consisting of Greeks, Albanians, Serbs, Romanians and Bulgarians. That failed of course.
There was also the school of thought of 'Hellenoturcism', a faction that believed that while being under Ottoman occupation was harsh, there was the mechanism to overcome it and eventually rule the Ottomans, like it happened with the Roman Empire where the Hellenic culture overcame the Roman dominance in the region. This was rather evident after the Greek Independence War as most people were still under Ottoman rule and there were lots of fans of this confederation or co-rule idea, which was farfetched at the time. Even in the early 20th century after all the conflicts, important Greek politicians still believed that it was a matter of decades but then the Balkan Wars happened following by the Greco-Turkish war that ended in 1923 so it's all 'what ifs' and wishful hypotheses based on pretty much nothing other than the fact that Greeks were rather successful in trade for at least a century (until they were heavily taxed) and there was some influence in the High Gate from the Fanariotes ('Fener' region of the City).
I haven't seen any fans of Hellenoturcism ever since the 1950s, maybe there are some still.
An Islamic Byzantine Empire would pretty much act like the Ottoman Empire I imagine.
There were various Sultans who did claim that they were the continuation of the Roman Empire. To a small extent they were right but that was purely coincidental.
