In Crusader Kings 2, to fully occupy enemy territory, you have to besiege all holdings. Usually, that is a castle, a town, and a bishopric.
I know that bishops sometimes led armies in the middle ages (badly defined as it is), but it seems absurd that there would be a "Battle of X Cathedral". Most of the clergy would be some combination of bookworm and lazy noble, so you wouldn't expect too many to be starting armed resistance groups. (I know this isn't a very academic way of thinking about it) If I were a Catholic leader of an army invading another Catholic area and I came across a cathedral for a major bishop, I wouldn't want to wage war against him. I would probably meet with him to help ease the transition. Plus, sweet! I have a new Cathedral.
All this to ask: Were there any sieges of Church property that were comparable to castle sieges or did the Church property generally pass peacefully (if politically) from conquered to conqueror?
If there were these sieges, how did that work? Were there soldiers stationed at the Cathedral to sound the alarm or something?
If it helps, in CK2, the current year for my Siege of Argyll is 870.
While there will always be more to be said, I wonder whether the following previous posts of mine might be of use to you:
The first, one of my recent posts, cites some cases based on written and archaeological evidence that the Vikings occupied the stone church and used as a part of their fortification for the siege. Christian army also sometimes attacked the church property and burned. or even took loot from it, and I'd say in some cases their violence did little differ from that committed by the Vikings. In other words, the stone church building / cathedral in Early Medieval West could sometimes become a focal point/ symbol of the fortification either/ both due to the accumulated wealth within it or/ and the strategic importance of the building.
On the other hand, the second and third post mainly concern the violence committed within Christian polities during the Viking Age, and local churches was not immune to this trend. Many episcopal and monastic office was occupied by those who came from the royal or aristocratic family, possibly educated similarly with their secular counterparts, and we sometimes know even better with the military mobilization within the ecclesiastical lands than other lands in Early Middle Ages.
So, I suppose that, regardless of how it was like CK2, 'the siege of X church' would sometimes occur, but probably not in every bishopric in Early Medieval West.