Did you check for power at the microwave plug.? Would hate to think something killed your microwave.
The wiring for inverters has a separate breakers for the circuits on the inverter. Mine does. A separate breaker box and it isn't even close to the main breaker box but on the ceiling in a forward compartment. It has two breakers, one for the microwave and one for all the other outlets receiving power from the inverter. If yours is similar, find those breakers to see if the microwave breaker tripped.
Pete - Full Timing
2000 Country Coach Magna 40', CAT C10, #5892
2006 Jeep Liberty