You may find that your generetor is not actually generating at 120 volts, it may be as low as 105 volts. This will cause appliances and motors draw more amps especially to start because they are striving to work at the rated wattage. Get a relatively accurate voltmeter, calibrate it by plugging it into your house outlet to see what the voltage is where you know the compressor starts and runs OK. Then check the running voltage on your generator with no load--nothing plugged in it--then see what the voltage does when you plug in the compressor. I betcha it drops down out of sight, maybe to 80 to 90 volts as it tries to start the compressor. The formula for wattage says that for the wattage to stay the same, amps have to increase if voltage decreases.
OK what to do about it. You said that your compressor is rated 120/240 volts. So it's super easy to rewire the compressor for 240 volts. If you run it at 240 volts the wattage of course stays the same but the amps DROP IN HALF. Problem solved. Just take the cover off the little plastic box that the line cord runs into on the compressor. There should be a little diagram showing how to change the wiring around with a couple of bayonet tap connectors. With my compressor I kept two different line cords one 120 volts and one 240 volts,The 240 volts oneI put a straight plug or whatever plug you need to plug into the generator 240 volt outlet. Compressors run better at 240 volts anyway if you need to use any kind of extention cord because you get less apparent voltage drop at 240 volts.
Synopsis: check your voltage and change the compressor to 240 volts.
simonmeridew