GFCI tripping on overload?
Overloads don't trip a GFCI. Period. If there was an overload, it would trip the actual circuit breaker. Since this is a GFCI/receptacle (as opposed to a GFCI/breaker) and the GFCI trips, that is a GFCI problem - 100% - and not an overcurrent situation.

14 AWG raises the question of the size of the breaker. If it is 15A then everything is good. If it is 20A then the breaker is oversized for the wires. If that is the case, the breaker should be changed to 15A. That is a general (fire) safety issue, unrelated to the GFCI issue. But even if it is 14 AWG on a 20A breaker with 18A flowing through (therefore over the allowed safe current for the wires but not enough to trip the breaker), it would cause overheating of the wire and potentially a fire - but would probably not trip the GFCI.

Now back to the real problem. Something is tripping the GFCI. Most likely that is some current leaking to ground. That is what GFCI is designed to protect. That also perfectly matches when humidity is high during the rainy season. When water gets in places in electrical devices that it shouldn't, electricity can move in unwanted pathways. It takes a lot of current to trip the regular breaker. It only takes a little to trip a GFCI or to injure or kill someone.

The good news is that it sounds like you already found the problem: "we disconnected one of the fans which stopped the tripping." If that's a consistent/reliable fix then the problem is either in that fan or the wiring going to it.

As far as startup loads: anything is possible. GFCI and AFCI both have sophisticated electronics. A GFCI should be relatively simple and immune to anything except actual current leakage, unlike AFCI which is trying to detect some relatively hard-to-detect problems and could be "confused" more easily by motors 'n things. However, newer GFCIs with automatic testing have more sophisticated electronics - and therefore are more susceptible to problems from other causes.