What size is the generator? Its output and motor starting capacity may well depend on the type of generator (Armature wound with rolled steel magnet frame or AVR controlled stator wound machine).
If it is only a 3/4 HP motor, it would be drawing about a kW at full power (assuming 67% efficiency). That might well be exceeded by 3 or 4 times at initial start up ,even on no load (inrush current at zero revs), so I would have thought it should start on your inverter supply.
Possibly your inverter shuts down as the initial surge exceeds its rating (some inverters are rated at surge, not continuous power) or your battery voltage drops too low, for a moment, and shuts down the system?
I can run my one horse power lathe from my chinese 2kW inverter with a 5kWh LiPo battery and I expect my 1.2HP motor will also start with no problem. But my motors are three phase and run through inverters, so they are programmed for soft starts.
220V @ 20A is only 4.4kW.
My guess is that it should work OK if you install a three phase motor with VFD. A 2.2kW inverter would work but the operating safety limits may not be programmable for such an over-sized inverter.