Mysophobe/mysophobie wouldn’t be understood at all by most if not all people in France.

Germophobe/germophobie would be better, but I guess some people would think these words are related to Germany.

A couple of words sometimes used, especially this year, are virusophobe and to a lesser extent virusophobie. They are more specialized than the former but immediately understood by everyone, although probably not considered as official, "serious" words. Otherwise, you can just say quelqu’un qui a la phobie des microbes.