You are tripping the GFCI because you have a ground fault in the RV and potentially a very dangerous situation. Get it fixed ASAP.
As to the battery, the power simply passes thru the converter. The fuses in the converter and a fuse or breaker at the battery are the only safeties in the system. There may also be a battery disconnect.
Was any work done recently? Any chance a wire was left disconnected? Battery replaced and the reverse polarity fuses in the converter burnt?
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