Typically Microwaves are 1000 devices. So you will need a larger than 1000 watt "inverter" to use it. 1000 watts at 90% efficiency is still about 100 amps of current draw. That is 1/0 guage wire if it is over 5 ft long to power the inverter. Energy is NOT free. Takes power to make power.

Sounds like one of two things are happening:

1. Microwave is drawing too much current, causing internal voltage monitoring circuitry in inveter to go into protection mode.

or

2. RFI from microwave is somehow being few into inverter causing similar circumstance.