I would say that they would switch out. When you're in a public room it would
display your public rank and when you're in a private room it would display
your private rank. Or if you don't like that then we could have it set to only
display your public rank in-game by your Top Right/Left name while playing and
in chat the mod rank that gets updated after each match would be your private
rank.

Something along those lines could work.