The Don Rowe inverter FAQ has a question about running an inverter off
of a microwave:

http://www.donrowe.com/inverters/inverter_faq.html

Unfortunately it's rather light on details. Anyway, it should work, but
you'll probably need a 2k running, 5k surge inverter to run it. They
are out there and actually not terribly expensive any more. Sinewave
units at that size are still pricey. A Xantrax 1750 for instance is
around $300 with the cable necessary to install it, and will run most
reasonably sized microwave ovens for as long as your batteries hold
out.

-Keith