Forget the battery generator thingy. That's nothing more than a (usually smallish) 12V battery and a (usually cheapish) MSW inverter in a single plastic box with a handle. You will get much better battery life from the trailer's house batteries, and be able to install a better inverter as well.
A 500W inverter would probably have trouble starting the compressor in the dorm fridge. Running it once started would not be a problem. The compressor might not like MSW power any too well.
(Don't be surprised if you find the mini fridge uses almost the same amount of power as your full-sized residential fridge. Most are not all that efficient comparatively speaking.)
