The 383TH has the GE residential refrigerator and 2000 watt inverter. The 382TH model has the RV gas-electric refrigerator.
First, here's the owner's manual for your refrigerator:
https://products.geappliances.com/Ma...-1000472-1.pdf
Page 43 lists reasons why the refrigerator does not operate:
Refrigerator in defrost cycle: Wait about 30 minutes for the defrost cycle to end.
Either or both controls set to OFF: Set the controls to a lower temperature setting.
Refrigerator is unplugged. Push the plug completely into the outlet
The fuse is bown/circuit breaker is tripped: Replace fuese or reset the breaker
Refrigerator is in showroom mode. Unplug the refrigerator and plug it back in.
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My guess is, the refrigerator is in showroom mode, meaning the light comes on but it does not cool. My Montana has a Samsung Residential refrigerator, and it requires a sequence of buttons to push to put it in Showroom Mode, or to take it out of Showroom Mode. This trips up a lot of folks with the Samsung. So, my guess is, it's in showroom mode. Nothing wrong with your camper or the refrigerator.
In order for the inverter to operate the refrigerator, you need to be unplugged from shore power.
The cut off switch between the batteries and the inverter needs to be flipped to the on position. This is a completely different switch than the battery "kill switch" which turns off the rest of the camper.
The inverter needs to be turned on. There is an on-off switch ON the inverter itself.
Then the refrigerator will run off the batteries, via the inverter.