Converter and GFI issues...and how does a Converter work?

The problem lies in your 120 V system and it may be that a wire is loose or shorted on the converter circuit. You can test the condition of your cord with a meter. I guess you already know it's not good.

Nothing necessarily wrong with using a standalone charger rather than your converter but a battery tender won't keep up with average use and your battery will eventually die. In fact, if your battery is severely compromised, the draw from the converter may be affecting things.