A question: How many BTUs is the A/C?
If it is below 13,500 BTUs, a Honda eu2000i can run it (although not it and the microwave at the same time.)
At 13,500 BTUs, it is iffish a 2400-3000 watt generator can run it, mainly due to the locked rotor amps. For this, I recommend two Honda 2000 units paired, or the above Yamaha 3000iSEB. Make sure to get the iSEB option with the battery boost so the generator can provide the additional 500 watts of inrush current.
At 15,000 BTUs, I'd definitely go for the Yamaha 3000iSEB or the Honda twins. Anything less, even "just" a 3000 watt generator won't hack it when the A/C starts up and demands its payment of 60 locked rotor amps.
As an alternative, you can use a hybrid inverter like Magnum Energy's or Victron's. This allows for a smaller genset, as the inverter and the battery bank eat the locked rotor amps.