I can run my 5500 BTU air conditioner AND my 1000 watt microwave oven simultaneously, with my Champion 2000 watt inverter! Caveats: The generator must be fully warmed up, the air conditioner must have cycled on first and then power up the microwave, and understand that the generator will be running in its highest RPM step if its in Economy mode, but yes the little thing will handle it.
I'll take a video if necessary for unbelievers, but it will have to be later after it warms up some around these parts, for the A/C compressor to cycle on.
