I have the same generator, except without the battery "boost" option. I have one trick I do which starts my 15KBTU A/C:
Stick a cup of water in your microwave. Nuke it for 30 seconds. Right before it is to cut off, flip your A/C on via your thermostat. 90-95% of the time, the generator is spinning fast enough to handle the compressor's inrush current and take the load. Once the generator is warmed up at the higher RPMs, subsequent compressor stops/starts shouldn't be an issue.
Oh, make sure the "econ-switch" is off. If the eco-throttle is on, the generator will not have enough RPMs to catch the inrush current.