I would suggest that you cannot unplug from shore power and unplug your battery and run the fridge on just gas. If your fridge is like mine, I need 12V to the fridge to run it on gas. With neither battery or shore power you won't have 12v to the fridge.
I would leave shore power connected and disconnect the battery to charge it.
Sure others will chime in with helpful advice too.
Chris
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