I decided to give it a try on the HTT and some of the time it really struggles and just can't get the A/C compressor to fire up. Seems that once it's going there isn't a problem even as the compressor kicks on and off, just on initial start up. If it just can't get it I let it run for a bit without the A/C on and after a while it does just fine. I am not sure the exact unit but the A/C is a Dometic and I am pretty sure it's a 13,500 BTU unit. Should this generator have any issues with this A/C unit or could there be another problem such as maybe with the generator itself?
The problem with initial start up is that the motor on the generator is fighting to catch up with a huge demand from Idle. It's like trying to tow your trailer up a steep grade and starting at the bottom from a dead stop. It's better to attack those hills from a higher speed at the bottom.
Same goes for your generator. You might try holding the throttle open by hand and have someone turn on the AC unit while the generator is running at a high RPM.
Starting your AC with the gen at idle is also hard on your compressor and will shorten the life of the AC because it is trying to start out at inefficient voltage.
The Honda Gens have a switch that kicks the RPM up for running higher demand items. It turns the RPM up to max (without over revving) and allows you to start the AC at max voltage.