I have an '07 27FB Classic and the microwave came installed in the middle slide out pantry section next to the cooktop. It is plugged into an outlet at the rear of the cabinet that has an interlock switch that makes the outlet hot only when the drawer is out. I have to wonder if a PO or a dealer replaced original microwave and relocated it? All the literature I've seen for the '07 27FB shows only the microwave in the slide out drawer.
Your description of the inverter behavior makes me think the inverter doesn't have the capacity to run that microwave. Mine didn't come with an inverter. It also sounds like a PO wanted a larger microwave and the ability to operate it without shore power and ended up with an unworkable combination.
Is there not a vent hood above the stove? If not, is there an external vent cover outside where the vent hood might have been?
Air Safe Hitch, Reese Dual Cam 800#, Centramatics