Paper taping or fiberglass mesh taping in bathroom
Could not we use... a primer and a moisture-resistant paint to protect the seams ?


YES, either tape will work fine and you should use the joint compound you are most comfortable with. When covered properly with a high quality primer and paint then the moisture should not be an issue for either.

P.S.- I do not use mesh tape because I have had experience with cracking that does not seem to happen as much with paper. Many folks will likely have suggestions and anecdotes both pro and con for both.