I'm at 182 dwellers and current living quarter arrangements give me max 200. The game put a cap on building more living quarters, which is absurd, since there's no cap on other rooms. Now I can't destroy any living quarters because I have too many dwellers, even though there's 20 in the wasteland and I'm sure they could do without a bed for 10 minutes.

So without murdering a ton of dwellers every time I want to move a living quarter how am I supposed to rearrange my rooms?

The only way to move quarters is to be under the amount of dwellers to destroy the room. Evict away.

It wouldn't be hard to change it so we can build as many living quarters as we like, but the limit will always be 200.