Stain is preferred, but I would wait for it to start peeling before you do anything...it may be okay, especially if there is never any direct sunlight.

Stain is pretty much just thin paint, so there's no reason they shouldn't be compatible. With regard to "cedar has to be stained so that it can "breathe"" I would say, succinctly, BOGUS. Cedar is routinely painted, has been for hundreds of years. There's nothing magical or different about it from other siding materials aside from some inherent rot resistance. It's dead, it stopped breathing a long time ago. It can be allowed to weather naturally more successfully that most other woods, but will last longer and (IMHO) look better with a coat of paint on it.

If you have peeling you have a moisture problem with the wall, not an issue with the coating.