Well, good descriptive info in your posting.
It almost sounds like the AC voltage dips and the fridge temporarily flips to propane and then the AC voltage comes back on/up causing it to take control again (being in Auto mode).
Normally a 20 amp feed to the coach should not be a problem (with some load control), unless maybe the 120vac is dipping out of spec briefly for some reason (Air conditioning compressor start up perhaps?).
One way to check if the AC issue is coach related somehow, is to use another extension cord from the house (different outlet if possible ; 15amp is fine) and connect only the fridge AC control board plug (in the outside fridge compartment) directly into the second extension cord. Assuming the house is not doing any brown out voltage dips, this should at least provide a known good power source to the fridge to see if it still has the flip flop problem. If it does, I would “guess” the control board may be flaky. Check all electrical connections on the control board for any loose or corroded connections first though.
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