Any PoD for this scenario must be before 795, as I said before. Pretty much the entire 8th century saw the ERE effectively in a case of civil war with iconoclasm. At the time, they were pretty much the only 'Orthodox' country that had any power. Both situations continued until the ERE sorted itself out around 860.
At the same time, the Franks, under Charlemagne or not, were becoming a very important force. Given the idea where Pippin III considered himself to be protector of the Pope, they were pretty much the carriers of a faith.
So the Avars pretty much have a choice of a strong and organised Catholicism, or a self-destructive and threatened Orthodoxy. Not much of a choice for any sensible ruler.
Somehow I find it more likely that they prefer to embrace Orthodox rite than the Catholic/Frankish one, if the relations between them and the Frankish realm remain hostile. Perhaps the Magyar invasions allow for a convenient alliance between Byzantium and the Avars, thus speeding up the Christianization process, or the classic "domino effect" of the conversion of the Balkanic peoples such as the Slavs and the Bulgarians. To make a comparison, I remember that the Hungarians themselves converted to Catholic Christianity in late 10th Century because they were wary of falling inside the resurgent Byzantine sphere of influence.
Or perhaps it happens the other way around: an eventual revival of Byzantium after they finally get ahead Iconoclasm might be hostile to the Avars, ensuring a greater approximation with the western Christian polities, like it happened to the Croats and the Carinthians.
The ERE at this time (790) didn't stretch much further north than modern Greece does. 'Avaria' is simply too far away for the ERE to worry about invading, especially in comparison to Bavaria (Catholic) which was right on their border. The conversions of Slavs and stuff has to wait for iconoclasm to go away - a minimum of 800 AD (if everyone lives roughly to the same time as OTL), and then the ERE was dealing with a civil war and Muslim attacks. They are simply too busy to worry about people that don't threaten them.
I'm not sure I completely buy this - the "resurgent" Byzantium under Basil and his immediate predecessors was never really that great a threat to Hungary (in the early 11th century they were effectively allies against the Bulgarians). The Germans were a much greater threat, and the conversion of the Magyar monarchy to the Latin Rite has always struck me as a means to deflect that potential aggression ("look, we're Latin Christians just like you") rather than as an anti-Byzantine measure.
My feeling is that whoever occupies Pannonia is more likely to be Latin than Greek for simple geographic reasons - it's too close to Germany and too far from Constantinople to make the Greek rite terribly attractive. I'm not saying that it can't happen, only that it would require a compelling justification or some very favorable coincidences.
The 11th century had Poland and Saxony and half a dozen other places Catholic, mostly because of Charlemagne (even though he didn't get to the Vistula, the effects of conversion had begun), so it made a lot of sense to join up with every other neighbour. In the 8th, everyone around them, except Bavaria, was pagan, and was going to be pagan for at least another century.
So, if the Avars have to become Christian, they will join the Latins. I think they will probably hold on to Paganism for at least until 900 though, in which case butterflies would have dramatically changed the world of the ATL.
- BNC