Here is my experience:
For a 15k BTU A/C, a 3000 watt Yamaha iSE (without battery boost) is iffish on starting. My trick is to start the genset (ensuring it isn't on Eco Throttle), let it warm up, run the microwave for 30 seconds, and just as the microwave is about to turn off, fire up the A/C. This way, it easily "catches" the inrush current from the compressor. Then, once running, all is good from there.
There is a battery "boost" model, but I'd rather do one of two solutions, were I doing it over again:
1: Best solution is an Onan with a generator controller. No having to walk outside to pull a rope, just push the start switch from inside. I have seen some very innovative Onan installs, including retrofitting a travel trailer with a 25 gallon gas tank and putting the genset underneath. However, this isn't at all possible in a lot of cases.
2: Two Honda eu2000i models that are parallel capable. Canadian models, and the GFCI models are not, so one needs to make sure that there are receptacles for the parallel wires. Plus, if one needs more than that, Wise Sales has a 3 generator parallel kit. Extended run tanks are available, because the 2000 watt generators have a diaphragm pump.
3: A larger generator with a remote start option. These are expensive and heavy, but if I were going with a generator for my current travel trailer's use again (which is going to change when I get around to things), I'd probably go with the 4500 watt Yamaha, or the Honda eu7000i, which has a dB rating as good as the 1000/2000 watt models. With remote start, it makes life relatively easier, and having horsepower to spare is nice, especially if one uses the variable speed setting so the generator can handle the power needed for an A/C at lower RPMs. An advantage of the bigger generator is the option to use it for the house if a disaster happens.