Overall, I think you have the bad gist about religion and politics in the Medieval times.
But wasnt the Church pretty much a 5th column? Bishops and their titles owed their power to Rome, not the feudal lord. If an upstart bishop wanted to play political shenanigans, there wasnt much said lords could do to stop them? Being the member of a different religion would negate that issue
That the Church was a 5th column was quite possibly true in the context of the Investiture Controversy, but not much anywhere else; most of the other time and places Bishops owed their power to Church politics that often mirrored the secular ones, being manned by the same noble families.
Outside of that issue, Bishops would at worst happen to endorse rival claimants, giving them more authority than a non-Church noble would, but still be bound by the widespread caesaropapist assumptions and unable to lay the claim to authority themselves.
You could simply revoke and liquidate his lands. I'm sure the Umayyads did that after the Iberian conquest
Obligatory "Pdox games are built different" here. The Muslims took over a lot of Christian land, but seldom outright stripped them of their properties or rights, rather preferring to levy an extra tax and restricting certain behaviors (doing processions, keeping wealth in churches); if only because more overt oppression also proved more dangerous. And besides, that Bishop probably is related to various nobles you still need to keep on your side, so liquidating him (and going through the trouble of finding competent administrators) probably bides poorly for your overall situation.