Possibly not even a Carolingian dynasty; the Papacy's role in legitimizing the overthrow of the Merovingian would probably be butterflied away with a Greek Pope instead of one trying to curry favor with the Franks. The imperial coronation was just the act that cemented a Papal-Frankish alliance that had been building up for some time.
Stephen II travelled to Paris to ask for help against the Langobard encroachment in Latium, and paid the bill by legitimazing the ascension to the throne of Pepin the Short. Now assume that instead of Stephen we have a Greek pope (say again someone from Sicily or Apulia or wherever in Byzantine Italy), His troubles with the Langobards would not change, and it is quite doubtful that the Byzantines can really give adequate support. The new pope has two possibilities: he makes a deal with the Langobard king or again he asks for the help of the Franks. The former might see the pope becoming a Langobard puppet over the next couple of generations, and might bring forward a strengthening and centralization of the kingdom of Italy. It would not wave away the risk of a war with the Franks, either in Italy (as it happened OTL) or as a continuation of the fight of the Franks against the Bavarian dukes, supported by (and intermarried with) the Langobards. My take is that this would not necessarily prevent (or even make more difficult) the ascension of Pepin to the throne (a bit like what happened with William the Conqueror: he would have gone on to England even without the papal banner). What it might however happen is that instead of going to Italy the eye of Pepin remain fixed on the Saxons in the north and the moslems in Septimania, and this might result in any kind of butterflies (maybe Pepin - or even his son Charles - die in some stupid little affray against the Saxons; or death come out of any kind of illness. Who knows?).
IMHO the big impact would be that the temporal power of the popes would be nipped in the bud: and this would certainly change the future history of Italy and Europe in a substantial way.
The alternative in which the Greek pope still convinces Pepin to help against the Langobards needs not to be discussed, I think: it would be very close to OTL again.
Seems likely without a seperate PoD; the main reason for the shift away from Greek popes was the inability of the Eastern Roman Empire to defend the Papacy from the Lombards and other threats in Italy, requiring the Papacy to find a new sponsor/protector. The best way to keep Greek popes in power is to keep a strong Roman presence in Italy.
Exactly what I was thinking. Probably the POD would be the Byzantines managing to stop the Langobard invasion and strengthening their hold on all of Italy. Sooner or later they will have to deal with the Franks, but that's another story.