I think you'll be fine with the generator as is. The fuel burn that you mentioned sounds like the owner was working the 2000i pretty hard. This generator is smart, and will automatically speed up or slow down according to the actual load being placed on it, and fuel consumption varies with the load. Honda's specs are readily available on the Internet and I'll break down what they mean for you (this should be close):
The 2000i is rated for a continuous output of 1600 watts, with a maximum output of 2000 watts for a short duration. At the rated load (1600watts) the unit will run for approximately 4 hours on a full tank of gas. The run time is approximately 15 hours at 1/4 load (400 watts).
With your smallish trailer, I assume (my wife Suzan always tells me don't assume) that your AC is a 5000 btu unit, which will draw about 550 watts when it is cooling at the max. You can check the power consumption on the manufacturer's placard on the side of the unit. Let's assume your small fridge draws about 150 watts. That would be a total draw of approximately 700 watts with both units working 100%. But, your AC compressor will be cycling on and off and the fridge will be doing the same, so the average power consumption will be something less...let's call it 500 watts just as a guess. If that's the case you should be able to run it for 12 hours or so without refueling...certainly long enough to sleep comfortably unless you're really hibernating.
Seems like someone on the forum had a picture of a gas cap converter that would allow for a remote tank to be used, though, if it did get to be a problem.
Both the AC and the refrigerator have compressor start up loads to take into account. I don't think you'll have a problem with this, but if both compressors want to kick in at the same time it might be a bit of a strain on the generator. You might want to leave the AC "off " and the refrigerator "on" while you start the generator. Then turn on the AC. This will stagger the start load. Once up and running I think the chances are slim that both units will cycle on at the same time. Perhaps someone on the forum has a 2000i, a window shaker, and a small fridge and can test this out for you.
BTW, Suzan is a finalist in this year's Genesis contest, romantic suspense. Hope you have a wonderful time on the book tour.