On my Miramar (2016) the inverter (2000 watt) runs the fridge and a few plugins on the passenger side, such as the TV outlets. The inverter doesn't operate the water pump or ceiling lights since they are DC current. The inverter just converts DC battery to AC current for the plugins I mentioned and the fridge.
With all that said, are your sure the inverter is turned on? I also believe there is a breaker on the inverter itself...speaking from memory so not sure.
In any case your residential fridge will be fine overnight from 10pm until 7am. I have dry camped without running the generator over night and bad batteries running the fridge...that's how I know. Just don't open the fridge door until you retstart the genny.
Hope this helps
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