How do I run in battle?

[caravel]

Yes, I seemed to remember that Bishops cannot request marriage, but wasn't sure. I think that may be the deal breaker as the player and AI will be effectively blocked from requesting marriage having to rely only of offers. It's not clear how well this would work.

Cardinals as Papacy only would add "flavour", but I'm not sure what it would do for gameplay. Without Inquisitors the catholic factions may need Cardinals as their second level zeal increasing agent.

Invisible Inquisitors are actually easier for the AI to deal with, as his border forts will mop them up automatically - if border forts were removed though, then I'd advise against it. Visible Inquisitors are impervious to counterspying and thus can often run about killing everything. It's often down to the player, who is better at raising, protecting and deploying high valour assassins to stop them.

Inquisitors in this form work well as they go about randomly trying random generals and starting mass inquisitions in provinces (reducing zeal). They work as a kind of "natural disaster". If they get caught, you'll never know they were there in the first place as they can show up as an assassin or spy getting caught.