The Honda EU2000i does power the 9500 BTU (+/-) A/C on our Casita. It also will power our Sharp micro/convect oven. BUT, not both at the same time. Early on it would not run the A/C. I learned that that was because our water heater would heat anytime AC power was on. There was a switch next to the water heater (accessed from outside). If I let the genie run for 20 minutes or so, the water heater shut off and it would then run the A/C. I chose to just turn off the AC (alternating current) switch and use propane to heat the water.
Another trick with A/C (air conditioning) is, if you know you will need it, turn it on as soon as you get up and start the genie. That way it will not have to work overtime to cool to the ambient temp. You will save fuel and stay cooler. ;)
Remember, though, turn off the A/C before using the micro.
LS