A 2000W generator usually runs at 1600W.
The 13500BTU Dometic AC unit draws 1670W of power, so generators that run at 1600W, but peak at 2000W will run harder, but will probably do it. They may not be able to start the compressor without something like an Easy Start.
I just did a trip in my new 176, in which I have an Easy Start installed. I used a Westinghouse 2200W generator, and it starts and runs the AC just fine. I had other problems with the generator having to do with the external 'extended run' fuel tank.
I am probably going to pick up one of the Harbor Freight Predator 3500W generators, mostly so that running the AC will only be slightly more than 50% output, leaving me overhead to run other (not so big) things. Part of that will be running a small ice maker, so I can remove the freezer box in the fridge, which should help the fridge get cold better.
If you want to use a generator to run your AC, take the above into account. The AC draws 1670W. You'll want a generator that runs at more than that. How much more will dictate how much more stuff you can run at the same time.
BTW, I also left my converter breaker off. My 100Ah AGM battery lasted just fine for three days powering the lights, fridge (on LP) and the thermostat. I didn't bother to hook up my solar panel. I don't think the battery went below about 70%.