Hi Ally, I imagine this is really stressful, I will try to help you out as I have 2 cats & 2 bunnies in my household. I don’t quite have this issue though, as I’m able to keep the door closed to the old cat & the new cat is used to the bunnies.
However, it gets hot during the summer & in order to keep my room cooler, I wanted to keep my door open. I still needed something in the doorway though 1) to keep my bird from flying outside the room, & 2) to keep the cat from coming in & the bunnies from going out.
First….
I made a gate out of NIC grids (you can get a baby gate though) & attach it like normal to the door frame.
I would make sure that kitty paws can’t fit through the gate holes, so a screened baby gate might work better.
If you’re afraid that the cat or bunny might tear the screen, You might be able to attach plexi glass or something in front of the screen, that way neither cat or bunny can scratch at it.
Second….
We know cats can jump pretty high (so can bunnies), to prevent this, you can get a magnet mesh screen to fit in the door frame much like THIS ONE. I used this screen to keep my bird from flying out. It allows the room to stay open & you can still pass through. It did get a little annoying after a while, so I have been considering getting an actual screen door with shades or something lol, a little odd but oh well 
Otherwise, I think your bunny can definitely get used to being moved to another area during night time. I would try to avoid leaving it in a small closet with the door closed though as it can get hot & stuffy. If it is a small closet, I would try to somehow make a 2nd barrier around the cage, fence it off somehow so that if the cat still tried to reach the bunny, the 2nd barrier would prevent it from being able to reach all the way. Hope one of these ideas helps or gives you an idea for something that might work better! 