I'd say it's possible, but difficult. As several have pointed out, much of what is now Greece was taken over by Slavic tribes in the 600s but was then re-Hellenized by the Byzantines when they retook the area in the early 800s. This was done partly by settling Greeks in the area, often from the descendants of Greek refugees that had fled to other areas. Southern Italy got a big Greek boost in the 600s from Greek refugees fleeing the Slavic invasions. It helped that some portions of Greece, like Attica, Corinth, and Thessaloniki, never fell. There was also active Hellenization efforts by the institution of Orthodox bishoprics and Byzantine themes.
Some key components why I think it succeeded and why Greece is now, well, Greek. The Slavic interlude was fairly short, with the Slavs there very underdeveloped politically. Said Slavs were constantly exposed to Greek culture and influence via the various Greek enclaves that survived during the period. So they were already 'primed' when the Byzantines reconquered the area. Then the process was allowed to continue uninterrupted for a long time with little interference. There were some Slavic revolts but those got put down pretty quickly, while Krum's & Simeon's attacks were focused on the Macedonia and Thrace area, not the Peloponnese. By the time there were serious inroads by a foreign power, first Tsar Samuel and later the Normans, the process was mostly complete, although there were Slavic enclaves such as those around Mount Taygetos that endured until the Ottoman period.
Using Hellas as a case study, I'd said a successful Hellenization of the remaining South Slavs (Bulgarians and Serbs in this case) would require a not-as-complete Byzantine collapse in the Balkans and an earlier recovery. Perhaps while Slavic tribes manage to break through and overrun the interior, the Byzantines retain control of more of the Danube river towns and 'Bulgarian' coastal cities such as Mesembria and Varna. This would probably require either the Bulgars falling flat or not getting going so much. That way Byzantine efforts to retake 'Bulgaria' don't face nearly as much opposition. Also fractured Slavic chiefdoms would be easier to assimilate than a large Bulgar Khanate or Bulgarian Tsardom.
One option might be to have the Umayyads decline earlier and longer, with more infighting and fracturing in the Caliphate, lessening the pressure on the Byzantines. And then have no Abbasid era revival. If the Dar-al-Islam fractures after only a century or two, things become much easier for the Byzantines. Give Constantine V free reign to focus on the Bulgars and maybe a few more years on the throne, particularly if it's against weaker TTL Bulgars, he could definitely break them.
If the Byzantines manage to retake Bulgaria before it becomes Bulgaria in the 700s or 800s, re-Hellenizing it is much easier than it would be after the OTL centuries-long slugging match that only ended with the poking out of eyeballs on a unprecedented scale.
And once the Bulgars are out of the way, Serbia is a much easier nut to crack. IOTL the Byzantines really couldn't do much pushing into Serbia without having a Bulgar/Bulgarian army come down and whack them.