Indeed, Bela-Alexios had presided over Orthodox church synods in 1166 with Manuel and his betrothed Maria, considering that there also was an Orthodox element in Hungary at this time, Bela would certainly have to convert to Orthodoxy, with a focus upon the "Unionist" attitude of Manuel towards the Papacy.
Another interesting detail; upon marching with an imperial army upon the Hungarian capital, would Bela reign from Constantinople with Manuel in order to seamlessly ascend to the Roman throne at his father-in-laws death? I cant see him ruling from Buda if there is an intention of him being Basileus due to Manuel having no son.
I actually would expect that rather than Buda, you could see Bela ruling from Belgrade, at least whilst he is Kaisar. Its on the edge of Roman and Hungarian territory, can travel to Constantinople and Buda via the Danube equally easily, and is secure from Roman OR Hungarian uprisings.
Plus, as it is closer to the Orthodox areas of Hungary, it allows him to focus on building support in that region, which means he could institute a form of the Theme system in Eastern Hungary.
Constantinople is great, but Belgrade is a good compromise for most circumstances.
But I'd be making sure there was a fortification in downriver Buda to make sure that loyal forces can hold the Hungarian capital and allow a fleet to arrive, supply the fortification and deploy from it to counter-siege any potential rebels.
It could even be called Alexionople, or Sandorvaros locally.