I feel like Greece has three paths to go on when defining its nationality:
There is the Hellenic path, which is the narrowest and the one Greece followed OTL and probably this TL. It primarily includes speakers of the Greek languages, with Orthodox Christianity as a secondary identity.
There is the Rumelian path, which tries to step into the footsteps of the Basileia Rhōmaiōn. It defines itself primarily by the Greek Orthodox Church, with Greek merely as a lingua franca for Rumelians. This is a more open path, including many Bulgarians and people in Anatolia, but it would also be viewed as backwards and medieval.
Then there is the Alexandrian path. This is the least likely in my opinion, but the most open to attracting new subjects to the Kingdom. This would be the presentation of the Greek state as a multiethnic, multireligious state based upon the tenets of Hellenic culture and the unifying presence of the Basileus. Basically Ottomanism, but from Athens. This state would lean very much on both the presence of the Roman Empire and the Hellenic influences throughout Anatolia and Rumelia. It would try to connect people's identities more with the ruins of ancient cities like Troy or the remains of bathhouses and amphitheaters throughout their cities to point to a shared heritage and culture. They would point out similarities in dress of the Bulgarians, Greeks, and Turks. It would try to define an "Aegean Race" that points to shared physiological features between the ethnicities. It's a hard sell, but stupider things have been accomplished in history.
EDIT: They can frame the Bulgarian and Turkish languages like many Britons do the English language. Their true heritage being that of Arthur and Boudicca, not that of Beowulf and the Saxons. Or at the very least, Beowulf is merely a part of a heritage dominated by Britons like Arthur.