Don,
A fuse will always blow around it's rated amperage. A circuit breaker 'should' trip within a few minutes if slightly overloaded, and few seconds if highly overloaded, but sometimes a circuit breaker will fuse together if the arking to open the contacts is extreme, and it will never trip again.
That is why I like to know that I have fuses when working with 480 volts, not a circuit breaker.
The auto reset circuit breakers are not all that precise. Manual trippable and manual reset circuit breaker like found on my work truck to power the Tommy Lift are higher quality devices, much more expensive than the typical RV stuff.
My guess? If you are in a RV park, perhaps the receptacle is wired backwards? That would explain both tripping the GFI and not allowing the converter / charger to work to keep your battery full while using your inverter.
My inverter is a Trace Marine inverter/charger, and it has a relay that bonds neutral to ground. If your inverter is like this one, it probably will trip a GFI.
My RV, when new, the portable GFI would trip if I plugged in the RV. SO I found that the battery charger was causing the RV to trip the GFI. Then I installed the inverter/charger, and could not use the portable GFI anymore.
Good luck, hopefully daylight will allow you to discover the problems!
Fred.