And if it's the process of crossing over that changes him, why didn't his personality change as well?

That's why I think there's still a mirror!Rimmer there. Physical form may change (as Lister could get pregnant in the alternate universe), but he's still our Rimmer. There must be the real mirror/Captain-ly Rimmer still (as Deb and Arlene were still there when Dave and Arn crossed in).

Why would exposing his deception help him any? It'd be exposed for him once Captain Rimmer showed up.