I don't think any of that new stuff is in any way related to the outlet GFCI tripping. A ground fault means that current that should be flowing in the neutral wire is going to the ground wire instead, i.e. a "ground fault".
The most common things in an RV that cause ground faults are the refrigerator heater element (if in 120vac or Auto mode) and an improper or missing ground on the inverter/charger. Could also be a stray strand or wire in the shore cord, though(was the plug ever replaced?), or an improper placement of a neutral or ground wire in the load center.
Ground faults can be very hard to find and most RV techs lack the electrical skills to isolate it. Even many electricians are frustrated by this kind of problem.
The suggestion to use an external charger instead of the onboard system is the quick & easy solution to keeping the batteries up. A Trik-L-Start won't do it, however, since it gets its 12v from the onboard charger anyway. The GFCI will still trip and no power to the TrikLStart. Just plug in a standalone charger and hook it to the battery bank..
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL