If you've never used your inverter it may simply be turned off. First this is a kill switch between the battery and the inverter that is a completely separate line than what feeds everything else on your camper. That switch has to be on. Second, the power button on the inverter itself has be turned on. The inverter won't turn on unless the battery line kill switch is on. Third, there is a transfer switch, farther down the line, that switches the power feed to the refrigerator from household electric or from the inverter.
When plugged into household electric, the electric lines feeds to the camper breaker panel first, then to that transfer switch. However, if that breaker is off, no power gets to the refrigerator, it should automatically switch over to the inverter powered by the battery. if the battery is disconnected, no power gets to the refrigerator at all.
My guess is, you've missed the first turn off switch between the battery and the inverter. If the inverter is powered up, then the problem is probably that transfer switch.
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2019 Montana High Country 375FL
2014 Chevy Silverado Duramax, 6.6L Dually