Some have said that this will work if you add a special start capacitor to your air conditioner. However, I think this may be marginal for this application. Honda 2000i is not rated for this; so if anyone has used this setup successfully, I'm sure they will add details to this thread.
Yamaha has a 2400 or 2800 (not very familiar with Yamaha) that several have reported they use with the air conditioner. This might be a better option, if you don't wish to parallel two generators. Or, you can go to a larger Honda 3000 or numerous other brands, although these are much heavier.
Personally, we have two Honda 2000's to run the air conditioner in the desert heat, and we have been very satisfied. In cooler seasons, we pack only one generator. If you do NOT have a heat pump and use the "optional" internal resistance heater in the air conditioner (installed as standard in our Bambi), one Honda 2000 will suffice in the winter.
If you decide to buy only one Honda 2000, I'd suggest you get the one with the 30 amp RV plug, since prices are always going up. I think the inverter has more capacity (even though it will not supply more current if not paralleled). This model costs only a little more, and you'll be ready to connect a second Honda 2000, if you decide to purchase one later.