Can I replace a standard outlet that's protected with an upstream GFCI to prevent nuisance trips?
If I were you, I'd remove the GFCI from the circuit (nut the wires together and make it separate, as in don't use the LOAD screws) and then put a separate GFCI in the other bathroom in the same way. It's expensive (and unnecessary) but you avoid having a bathroom outlet inexplicably going dead (trip the GFCI bathroom outlet directly and your normal outlet goes out). This way, each GFCI can trip independently of the other.

As to nuisance trips, assuming you have no wiring issues, the GFCI might be going bad. Replacing it isn't a bad idea.