Short answer: NO
Firstly, two batteries will not run your microwave long enough to make but a couple batches of popcorn. (Just did the math: at best the microwave will run for around 20 minutes and completely kill the batteries.)
Secondly the generator is wired to power the battery charger along with everything else on the 12 volt system. Taking power from the battery to run the inverter to power the battery charger is insane. Separating the 12 and 120 volt systems is not as simple a task as it appears.
When you want to use the microwave, hook up shore power or buy a generator. The cheap shortcut of just adding an inverter is neither cheap nor effective.