Plugged into shore power or plugged into a generator should be equivalent since 120v is 120v. You are running the same 120v electric heating element in both cases. It's likely outside factors that cause the descrepency you are seeing. Ambient temps, warm items close to temp sensor, etc. Put something warm next to that sensor and you'll keep it going until the offending object cools down while freezing other things inside.
Ive installed some fans in my flue stack and on the interior fins but my fridge runs a little better on propane than electric. I need to watch setting it at 4 because I'll start freezing things.
2009 Georgetown 350TS (bunks)
400W solar, 440AH 6V GC2
2009 Ford Edge AWD Ltd towed
2011 Honda Fit Sport towed