It's rare to find a 2000 Watt generator capable of starting a 13,500 BTU A/C unit. Just not enough surge capacity. Do a search in the Motorhome magazine forum, this topic has been talked about often. Honda 2KW owners say no go.
The Honda 2KW "can" run a roof top A/C unit if you bridge it with another 2KW. But you need an interface cable. You'll be over $2000 by the time you're done. The nice part each generator weighs about 50 lbs.
The next step is the Yamaha EF2400iS. It's designed to run a 13,500 BTU A/C unit. It's actually a tad quieter than the Honda 2KW. Weight is 70 lbs. Good grab handles. Price for me was $1147, then received $100 rebate through Yamaha.
The Honda 3 KW is almost double the weight of my Yahama. Now getting beyond what can safely be picked up by 1 person!
Kipor generators are slightly less money, but are Chinese knock-off generators. Good luck ever finding a service center or parts. You may save a couple hundred bucks up front, but you'll likely lose more in the long run. Think Harbor Freight (also Chinese knock-offs) tools are as good as name brand? If so, get a Kipor.
I've done some testing with my generator. Details and pics are here: http://www.2manytoyz.com/yamaha2400.html
And if later you run into a bucket of money and decide you need more power, the Yamaha can be bridged just like the Honda.