I guess this is rewording what you have already said if I got it right.

Since the problem you see is in one configuration, I'll forget about the rest for now.

The inverter is the source with its one GFI outlet.
The double male plug goes between inverter output and power source intended for charger.
Unless there is an AC panel breaker for that charger circuit, then you are connected to the shore power input side of your AC system. (If there is a breaker for the charger circuit you'd probably have to turn it on).
It sounds like your charger outlet is wired in parallel to the input of your shore power panel.
Is the GFCI on the shore power panel on the master breaker or on a breaker that energizes your boat's multiple AC outlets?

All of these questions are only to get at the troubleshooting sequence to isolate the ground fault. It sounds like every bit of your AC wiring on the boat hangs off the inverter when you have it running.

I suppose you could have a little leakage here, a little there, but what the GFCI sees is enough to trip it. Or maybe its only one item.

I'm not sure it would be a factor, but do you have a galvanic isolator installed?

Sounds like you might want to try temporarily disconnecting part of the AC wiring a bit at a time until you have isolated the source.

Apparently this scheme has worked fine in the past, so has anything been changed in between when it worked and when it didn't?

If the only thing that changed is the passage of a winter, I'd be suspicious that salt, dirt, and moisture might have provided the leakage path to trigger the GFCI.

As my old physics prof said, keep one hand behind you when working with high voltage. Oh, and don't stand in a puddle of water. Good luck and hope this helps you think it through.

I assume that you are testing without the shore power wire connected at the boat end. Do you have a regular male shore power connector on the outside of your cabin side? If so, that sounds like another point that could shock someone when the inverter is on. Even though you have warning signs everywhere, it still is a little spooky to me. Back to the original thought about the shore power connector. They sometimes get cruddy and could make a leakage path.

I can't wait to hear the results!