Your battery will carry the load for microwave use and the long run from the engine to the trailer battery would not carry the load. Meaning the light wire and length would not supply the full output of the alternator even running full tilt. Leaving it idle should be ok but there is a chance you may pop the charge line fuse.
Just run your microwave for lunch, then drive away while the battery refills. Or if you have a mind to you could mount an inverter under the hood (lots of room ha ha), and then the full alternator output would be available.
