It appears to me, the issue of the generators is primarily due to the wrong A/C installed on the TC. From what I have read, most TC manufactures install A/Cs that are way to big on the TCs. A 7000 10000 BTU A/C is more than enough for the TCs. Bigger than that, and they loose a lot of efficiency. There have been many threads/discussions on this. It is my understanding you can have soft start kits installed on the oversize A/C, but my preference is the proper sized A/C.
I think a 2000 watt generator will efficiently run any of the 7000 10000 BTU A/Cs. My point here is I would insist the manufacture install the correct A/C, regardless of my generator choice.For generator choice, I think it depends a lot on use of the generator, how much you are willing to spend, and your mechanical ability. If you are a do it yourself person, I would not want a builtin generator. Just about any town will have a repair shop for the Honda, labor or parts. Is this true with a built in? If you want to use propane instead of gasoline, there are conversion kit that allow ether fuel.
Definitely an advantage to a built in, but for me the disadvantages fare outweigh the advantages. My Honda 2000 runs my 9200 BTU A/C (in ECO mode) with no problem. I can get about 8-hours on one tank (1-gallon) of gas. With my extender setup, I can connect a 5-gallon can. I have a mount for my Honda and extra fuel. I do have to pull a rope. This has worked well for me since 2005.
You can see the Honda 2000 below the red Boat fuel tank.
