In my experience cats living together pick a couple of spots that are "theirs" so even if they are in the same room, they have their own territory. By shutting the door at night you are keeping him from a place he identifies as his, as his safe and happy place, and that is causing him anxiety.
My personal option would be to figure out a way to screen the window so he can be in the room - it's possible you could ask your landlord to screen the window (one landlord in my past was willing to make even more expensive changes for me, as a good tenant), or you could install a temporary one pretty cheaply (I've done so for less than $10 per window). However, I actually like sleeping with my cats, and you've said you don't want to do that.
So what you need to do is to help your cat have different, and better, places that are "his." I agree with denying him access to your bedroom all the time. Also important is to deny the other cat access as well, because letting the one in but not the other may well just make him anxious and stressed. And then, create some lovely comfy spots for him, like the blanket under a dresser you mentioned. Use treats or catnip to encourage him to go to those spots. When he's there, pet him a lot, cuddle and love him so that becomes a "loving spot." It may take time, but he can develop a strong preference for a new spot with encouragement.
Feliway works for many cats to ease anxiety and stress issues. It's worth a try before going to prescription medicine. It is not particularly cheap, though - between $15.00 - $20.00 per month for refills per plug, and depending on the size of the apartment you may need two or three.
I hope you can figure something out - it's no fun when you can't sleep, and I'm sure the kitty isn't very happy either!