You guys danced all around the "starting" current issue without EXACTLY hitting the nail on the head !
ALL generators have a hard time starting a heavy load like an air conditioner. A large starting capacitor
(as used on conventional A/C units) helps but it also hurts if you have limited "instantaneous" power available
(generator). Inverter A/C units also have large capacitors. An uncharged, large capacitor "looks like" a short circuit which is why the "in rush"/starting current is so high.
What is required is a device that limits the current charging a completely discharged capacitor. The capacitor will still charge up but it will take 100s of msecs instead of <10msecs. Of course, the device has to "get out of the way" once the capacitor is charged up ! Because inverter A/C already have a lot of electronics on board, adding such a device when the product was being designed would only cost a few dollars. Some have this, some do not.
This company makes one
MicroAir EasyStart soft starter
If you want more gory details on what I said watch this video
How to make a soft starter