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Start by having the batteries load tested to be sure they are still good. Most of the appliances are controlled by 12 VDC power. Also get a small digital multimeter and we can check the charging system to be sure it is working. Make and model of your inverter would help and make and model of your RV so I know what we are working with here.
Did you check the battery disconnect solenoid switches to see if the batteries are even in the loop? You will need a small digital multimeter for some testing here so we can find out where the problem is. About $20 or so if you don't have one and easy to use.
Just checking back to see if you were able to resolve this issue or do you still need some additional input from me here?
It might be, but it can be the automatic transfer switch as well. Testing would be able to tell you that in a few minutes with a meter.
By taking the cover off and using a meter to test the input and output of each side. The microwave gets power from the same source as the rest of the unit.
Since I didn't hear back from you I am wondering if you were able to resolve this issue or do you still need some additional input from me here?