If you can't get higher current out of the inverter you might try lowering the surge current of the load.
If you can get to the start and run caps for the compressor you might try reducing the value of the start cap. That would lower the initial start up current on the compressor but if it's too low you could fry it.
The compressor will spin up slower but with less surge current. I'd start with a value 2/3 of what's in there now but wouldn't go less than half. Make sure the caps are non polarized and have at least the same voltage rating. If the compresor just stalls turn it off immediately! Any electric motor repair shop should have plenty of them in stock.
Another thought is to put a switch on the fan motor and let the compressor fire up first, then turn the fan on. A timed relay or current sensing switch, other possibilities depending on how your unit is wired but a simple hand operated switch will tell you if it will work.
The small AC unit I have draws about 500W (4.7A@110V according to an Amp clamp) while running but when I first turn it on it can kick the 15A breaker if there's anything else on the circuit. Old house, half the outlets in the place are on one breaker including the ceiling lights in all the rooms. Leave 2 lights on and and start the AC I got to run out to the garage in the dark to reset the breaker. Cheap rental, I'm not going to rewire it.
Both motors are trying to start up at the same time and I leave to "power saver" off so the fan runs all the time so I don't get the higher surge current.
HTH
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