I used to have a Yamaha 2400 generator, and it would run a 13.5K A/C, but just barely. And then I couldn't run much else. I think 3000 watts would be okay, so maybe check into quiet generators at that power level. I've heard good things about the Champions for the price.
We have the twin Honda 2000's, and there's a reason why this is such a popular combination. Enough to power our A/C plus minor other stuff; it's much more convenient to move two small generators than one large one; QUIET!; the flexibility to only run one when small power is needed. I also have the extended run tank that holds 5 gallons.