the inverter receives 12v power from the batteries, because they are 12v output, and the inverter changes that to 120v output - the Volts 'reading' that we are referring to is showing the amount of the battery bank 'storage' capacity left.
On one of our first major trips, overnighting during freezing temps, I didn't know too much about the Volts reading. Everything worked fine until the Inverter eventually cut out due to the LBCO setting, which the factory had defaulted to 10.0.
Well, as you might expect, this was then not enough battery capacity left to start the generator!
I learned eventually that 11.5 is just a good number - not too low, not too high, at least for our needs.
13.5 might show when the Charger is fully charging the battery bank, 12.8 might show when not, and very little, if any load is on, 12.2 while you have items in use, 11.8 while items have been use for some time, 11.5 might be a little 'low', as it represents 50% or less of the battery bank's storage capacity.
all that said, I think you'll need to do some more research on the fridge, itself, as I don't think you have any issue with your Inverter, or it's output, especially if everything else works fine ...such as microwave, and/or other devices plugged into outlets that are served by the inverter.
or, if the fridge's outlet is the only one served by the inverter, plug something else into it, or the rear/front of the inverter, if it has it's own outlet, and see how they work.
none of this may be the ultimate solution, but at least give it a try
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