I haven't read all the replies here but I can tell you the short answer to your question is "No". I've spent many a weekend on generator power as we used to use our trailer as a "pit" trailer for off road motorcycle racing. These races are conducted on private ranch land where the only power, shade, water, etc is what you bring with you. My last trailer, as well as our 07 27'FB had a 13.5 A/C unit, plus microwave, and all the amenities. I finally settled on a 5,500 watt OHV generator. Yes it was big and heavy, but it did the job without a hiccup. On those 105 degree weekends in Texas we ran the A/C on the generator 24 hours a day and it didn't seem to notice or take a dip when the microwave was in use while the A/C was on. A smaller unit, even a 5,000 watter, sometimes tripped the breaker when we used the microwave at the same time as the A/C. So, all in all, I'd suggest that you get as much power as you can lift into the back of your truck.
You'll be happy with the results. Next time around I'd go with a Honda unit, since they are so much quieter than most others. These days, however, I'm using a 20 year old Kawasaki 1400 watt that I bought from a friend. No it won't run the A/C, but it will run the lights and microwave and other 120 V amenities on board. Since retiring from the bike racing we usually stick to the camp areas where hookups are available during the hot season. But in the cooler months the little 1400 watter does the trick, plus it's light weight and packs away easily.
If it ever dies, I'll replace it with one of the newer Honda 2000 watters. those are the way to go if that's the power range your looking for, plus it can double up with a 2nd 2000 watt unit for a double load. Use one when you don't need both. Use both when you need the extra power. They're light weight and easy to transport too.
Fuel for Thought- hope it helps.