The DC side should have nothing to do with GFCI compatibility. A bad battery or short in the DC side may cause a draw that exceeds the converter's capability and that may cause the converter to fail or a circuit breaker to trip, either a DC breaker or AC.
An easy check to see if the converter is functioning is to measure voltage across the battery terminals when the converter is powered. A voltage of at least 13.5 should be present.
I'm with Scott G's diagnosis--Not knowing the exact history of the converter, the most likely situation for a GFCI trip is that the hot and neutral are reversed.