ktmrsf, Good explanation but I suspect that the OPs Class A MH has a built in gen that has its neutral and ground bonded.
Check for leakage between the shore plug and ATS. Or possibly the gen voltage or it's waveform is different enough that the GFCI doesn't detect leakage with the gen.
GFCI's do fail and replacement can be a simple solution. Just be sure to use one that is compatible with the alternator and gen if they specs a certain GFCI.
Replacing the ice maker may also solve the problem. I'd certainly use an extension cord from the ice maker to several different 20A GFCI shore plugs to see if it by itself trips the GFCI.
Updated compatibility to include the alternator.