What make/model of A/C do you have?
My only concern is right off of wisesales power usage spreadsheet: GeneratorSizing
I too have a 12k BTU A/C, and notice the starting wattage vs maintaining wattage is north of 2000. If I get that Honda, it would stress out the little guy unless I had two running with the connector cables...and like Cam said earlier...it may be too much.
Can you let us know what happens when your A/C kicks in to the Honda? Does it shake, drop or rev up? Does it shake? Conch out...?