You keep mentioning the generator struggling to start it, especially in eco mode. Sounds like a small inverter generator, while the a/c is rated for 9 amps running its rated at 45 amps lock rotor startup current. That isn't bad, my 15k btu is 70 amps startup.
Most small generators can't even start a rooftop air at least in 13k btu and up even though they can supply enough running amps. Eco mode at least on my Honda 2000 throttles down to reduce fuel consumption then will react to heavier load by throttling up but it has enough delay that a large startup surge from an A/C makes it struggle even worse.
If you want to resolve the compressor start issues the best way is a soft starter like a Micro Air easy start, it greatly reduces startup surge. I added one and now I can start my 15k btu off my inverter.
As far as running amps being high, as others mentioned could be something up with the compressor, or dirty coils, or low voltage. As voltage goes down amp draw will go up as the compressor needs the same watts either way based on conditions.