OMG - tracking it down to a staple!
I just talked to a trailer electrical repair guy and he felt like it was no big deal - that some converters simply don't play nicely with GFI's. Not sure that I'm fully comfortable with that answer. It seems (from what I've read on these forums), opinion is split on this issue, with half agreeing with the repair guys' opinion, and half saying whenever your tripping a GFI its a safety issue that should be tracked down and corrected.
Everything works (AC, microwave, battery charging, etc) if I use a cheater plug (without a ground), and I suspect that everything will work fine on a non-GFI outlet. It sounds like many areas are going to GFI only, and the cheater plug seems like a bad idea in general.
Other than disconnecting the converter from the battery, are there other ways to make sure the problem is within the converter itself. Can problems on the DC circuits cause this problem? (such as a staple hitting a DC line, bad water heater, etc)?