Wouldn’t it be easier to just keep the door to the room shut so the cat can’t get in there? It doesn’t sound like the cat has lived with the OP its whole life and with the expectation that (the OP’s) bedrooms are okay to be in. I have no cat+bed problems primarily because cats are not invited into bedrooms in this house.
I don’t have cat pee issues but I also have no bedroom door. The steps lead up and into the open bedroom.
There are a lot of times I wish there was a door though. I love my pets but there are times I wish I could escape from them in my own home. I am lucky to go to the bathroom with out company.