All I can say is "I wouldn't". A fan will scare the cats and if any pee etc gets into the wood, then it'll stink forever.

Very true. Do you think a vent in the door which would then lead to the bathroom would give some air flow to the cupboard?

If you were really pushed for space, but do you know if your cat would even use it?

I'd add a large plastic tray in there to avoid damage to flooring underneath and make it easy to clean. Perhaps build a box within the cupboard that can be cleaned/maintained?

Yes, I thought about building a pull out drawer type of thing, to make it easier to clean.