You probably are looking at the start capacitor, not the run capacitor. Most motors will draw 3-5 times their running amps at start up, but only for an instant. "Soft start" motors are becoming more common, but are complicated and expensive. The reason you do not blow your breaker is that motor breakers are of the 'time-delay" variety and will tolerate the starting amp draw for an instant. Sustained high amperage will blow the breaker as designed.
If it is any consolation to you, purchasers of house-hold generators are discouraged by the size gen-set they need to rock over their a/c or, more importantly, their water well pump. The fundamental problem lies with the gen-set's inability to speed up, raising it's output, in the timely fashion (read instantaneously) to supply the high amps needed for start-up.
It is possible that running the a/c off a rather large inverter, with the gen-set charging the batteries, could work. This would presuppose a large battery bank. It would also be rather inefficient and shorten the life-span of your battery bank.
A viscous clutch, between motor and compressor, might allow the motor to start in an unloaded condition but the fabrication cost would make the gen-set and a/c set seem cheap.
If the generator has a manual throttle you might attempt revving it and then attempt start-up of the a/c unit-probably a two man operation. I continue to think that a conversation with your local electric motor shop, and there's usually a couple in decent size cities, could shed some light, if not amperage.(g)
“Scientists are people who build the Brooklyn Bridge and then buy it.”
Wm. F. Buckley, Jr.