I agree that 1.1 max amps sounds way, way low, especially for startup of the compressor. I would disregard the label. It might be an error, or they might not be talking about startup amps, or who knows what.
A GFCI outlet trips when it detects current leakage in the downstream circuit. If for example 1.1 amps flows out the hot wire and 1.05 amps flows back through the neutral wire, the GFCI assumes the difference is leaking to ground and might be going through your body.
It also could be a bad GFCI on the inverter, but that begs the question of why it started tripping immediately after changing out the refrigerator - if that's when it started. While changing the refrigerator, was anything else in the electrical supply path touched?
I suppose it might also be possible that your 3 year old batteries are worn out and aren't delivering as much power as you need, and perhaps the GFCI circuitry is failing before the inverter throws a low battery error. Original RV batteries on a 2014 rig probably need to be replaced. You can get them load tested at most auto shops or auto parts stores.
I'd start with the battery testing due to their age. If you replace them it won't be a waste of money even if it doesn't help with this problem.
If the problem continues, for about $25 you can purchase a power strip with built in GFCI. Remove the plug that goes into the inverter's GFCI outlet and plug into the power strip. Plug the power strip into an outside power source. With shore power off, see if that resolves the problem. If so, you have a problem with the inverter.
If the GFCI on the power strip trips, something at the transfer switch or refrigerator outlet, or with the new fridge is causing the GFCI trip.