Yes I'm not sure what impact lights would have on microwave use. I agree because they are 12 volt impact should be minimal.
It is a 1500 watt microwave and a 2000 watt inverter. I figured the inverter will have some loss and not deliver full 2000 watts.
Also the inverter energizes TV circuits, although TV's are not on, there is some loss there.
It's obvious I'm on the edge of the inverter ratings. Not trying to bake a cake. But i could heat up leftovers along the roadside I'd be a happy camper
So, it would appear the specs work out and the inverter should be able to run the microwave.
Having said that it might be time to look elsewhere.
1. Is the 12 volt battery healthy? Can you load test it? Are the cells filled if a lead acid flooded type?
2. What is the 12 volt battery rating, does it have the necessary capacity?
3. Are the inverter input and output cable connections clean and tight?(Battery connection as well)
4. Are the inverter input and output cables the proper sizes based on load and length to prevent voltage drop?
5. Are transfer switch connections clean and tight?
I am making the assumption you have no issues running the microwave on Shore power.