Can't rearrange vault because of Living Quarters/dweller limit

As the game currently is, there is no way to manage this, without getting rid of your dwellers first.

The main issue is that the only way to allow yourself to remove/rebuild the living quarters is to reduce the population. This means either killing (sending them to scavenge until they die) or evicting them, and yes, you will have to do this for enough dwellers to reduce the total population by the amount that one Living Quarters holds - i.e. 40.

I am yet to test this, but it may be possible to reduce this problem in future, by splitting up your Living quarters into individual rooms (rather than combining them). This will mean more rooms in your vault in total, but the population control is far less severe the next time you wish to rebuild.

I don't believe the game supports this directly.

You may have some luck using a save editor, depending on the platform on which you play the game. I have in the past used Fallout Shelter Save Editor with success to re-arrange my vault. I have used this on PC to edit a save from the Steam version of the game. It claims to work on Android as well.

There are other save editors reachable via google, but I have no experience to back up the claims they make about their capabilities.