Since the OP has already purchased a generator now, this info is for other perspective buyers.
A Honda 2KW is a tad too small to reliably run a 13,500 BTU A/C unit. It "might" start it initially, but probably won't restart the A/C unit after a head pressure has built up. Most report it won't even initially start theirs, including my local buddy here in flat FL. Two certainly will, but that's $2K worth of hardware with the parallel cable.
The Yamaha EF2400iS is the smallest, lightest, quiet series generator, that will reliably run most 13,500 BTU A/C units.
It weighs in at 70 lbs. A 3000 Watt generator typically weighs double that.
The 2400 uses the same sized engine as the 2800 and 3000 Watt generators.
It will carry a 3000 Watt surge for a full 10 seconds before going into overload.
I've done quite a bit of testing of my Yamaha. Details and pics at this link: http://www.2manytoyz.com/yamaha2400.html
Every 3 dB is twice the sound energy. Every 10 dB is perceived to be TWICE as loud to the human ear. Quiet is expensive, loud is cheap. Loud can't be fixed with a different muffler (doesn't address intake and mechanical noise). Pay once, cry once. I didn't, I cried twice. But I'm finally happy now!
Another thing to consider is when comparing noise figures, everyone looks at the top figure and compares apples to apples. Not so. Inverter type generators only run as fast as the load demands. A 5000 BTU A/C unit keeps my generator at idle, or 53 dB. My 13,500 BTU A/C unit brings it up to about 2/3 throttle, or 55-56 dB. A conventional generator is running 3600 RPMs constantly just to make 60 Hz.
If you can live with loud, and camp in the boonies, go for cheap. Contractor grade generators will certainly run your camper. I did this for many years. But if at a campground, or for emergency power at home, there's a HUGE difference in the sound level of a quiet series generator.