I've had the same idea earlier this year, but I abandoned it since I'd have had to follow Byzantium's and Hungary's history at the same time.
My PoD was that Manuel I is struck by a disease and dies at the end of 1168. His heir at this time is prince Béla (the would-be Béla III of Hungary in OTL) and his wife is Maria Komnenos.
He adopted a greek name OTL - Alexios - so he is crowned Byzantine Emperor as Alexios II. Before his ascension he felt that converting to Orthodox faith would secure his rule.
He intends to be in good relations with his homeland so he gives Dalmatia back to the King of Hungary, Stephen III, who was his older brother.
After this, IMHO, he can focus on the east.
In 1172 OTL, Stephen was poisoned and had no legal heir.
Prince Géza would become the new king unless Béla/Alexios decides to lay claim on the throne, resulting in a war of succession.
OR
Maybe Stephen's devotees poisoned his brother so he may assume the Hungarian throne after being denied the Byzantine one. But since he is no longer Catholic, he won't be accepted as a legitimate ruler. Or even if an archbishop crowns him, the Pope would be quite angry. So his followers no longer have motives to posion the Hungarian king, thus he stays alive and the good relations between the two brothers and states continue to exist.