Is there a 4x where religion is a big deal?
Castles 2: Siege and Conquest, an old DOS 4x, had the papacy as a unique side-faction. It was interesting, kinda a force for peace and tempering expansion. Basically, if you conquered territory from a faction which was in good standing with the church, the church got more and more upset with you. This had to be mitigated by tithing resources, or even ceding entire counties to it.
The ceding had some tactical possibilities, too - enemy lord kicking your ass, but you only share a single border? Cede that province, and it essentially became a wall no one dares attack. Plus the church loves you, and starts frowning at the other guy.
Why care what the church thinks, though? Well three reasons:
1) The papacy is the only faction that's usually at peace, so it quickly grows an absurdly large army. Particularly since everyone is tithing it resources. Fortunately it almost ever uses these defensively.
The papacy's opinion of you is a large portion of your "right to rule". You don't have to conquer the whole map, just get that score high enough and claim the kingdom - the church actually crowns you.
If you let your standing fall too low, you become excommunicated. Other lords refuse to negotiate with you, you're fair game for conquest, and your *own people* suddenly plummet to 0 morale. It's rough.
... But it's actually survivable, and it only happens once. If you pick a good time to do it, and spam festivals for a few months, you can actually recover and be free of the church's influence. Essentially an anti-pope situation I guess, though it doesn't explicitly say that. You're still at war with everybody forever, but it can work in the end-game.