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You have blown the microwave internal fuse. If you are handy enough you can take the cover off and follow the power cord in the system. The fuse will be between the transformer and the power cord YOU MUST DISCONNECT POWER CORD AS PART OF ELECTRICAL SAFETY WHILE WORKINGON ON THE SYSTEM!!
Look at the rating sticker on the oven and use the AC power rating as your guide. The cooking power of the oven is not important, but the power it needs from the AC supply is. Voltage times current is the supply power needed (watts), and you want an oven that will draw less than the inverter's continuous rating, less any power being used by other appliances.
For instance, a microwave rated for 120 volts AC at 10 amps needs 1200 watts. Your inverter would be OK with this as long as other devices aren't using more than another 1000 watts. You can't use the inverter's peak rating. That's only for brief bursts, like when the microwave first kicks on and draws some extra current.
Any mid-power (600 watts or so) oven should be OK.