recommendation for generator

I point the exhaust out towards the open tailgate and the lift window, and the sliding side windows are usually open part way.

Of course this works well to power the trailer when I really need the A/C or microwave.

It also reduces the noise levels to the side and front of the truck...which is nice if you have neighbors in the adjacent campsites.

One problem tho, is that crawling in there to pull the starter cord, is a bit of a pain and you have to get out pretty quick once the genset is running. Of course a genset with wireless remote start would eliminate that problem.

Can you slide the genny onto the tailgate and direct the exhaust away from the truck, trailer, and your neighbors? Of course taking steps to be sure it doesn't walk off as TT warned. 

Depending on how much tree cover there is where you camp, if you're not running a/c, your microwave or other big ac loads, a quieter, lighter, and less messy solution to consider might be to get a portable solar kit. That plus a dual battery bank seems to meet most people's 12V electrical requirements . You can always charge a bit from your tow vehicle via the 7 way connector if you run into a long string of cloudy days. 

We normally don’t run the AC. I think we have run it twice while camping since we have owned it. Most of the time we are outside, anyway. 

2 considerations:

1st. Carbon monoxide. Having the gate and windows open should help prevent this.

2nd: Heat build-up. If the exhaust is aimed toward the open tailgate, then the airflow should be sufficient.

Why not run the AC? That generator is large enough and if the day is really hot, you might appreciate it.