Generally the 12V output on cheap generators is limited to 8 amps or so, and is often unregulated (that means the actual output voltage changes with generator speed, is likely electrically noisy, etc.). The point is, even if you can deal with dc-dc issues, you probably only have about 100 watts total available on that outlet. Not enough for the scenario you describe.
It's best to use the UPS as designed. Which is to run your equipment long enough to allow for a clean shutdown if 120V isn't going to be available soon.
On the other hand, the refrigerator is a cold bank. If the situation demands, you could get away with not running it at all for a several hours (if nobody opens it), then running it for a few hours to cool down. How long depends on the insulation of the unit in question and the thermal mass of its contents.
If you have to choose, well, just live without your gaming machine for a while in such a situation. Putz around on your cell phone or laptop.