Looking at the motor its almost not surprising that you shut the generator down.
You are just shy of 4000watts running full load
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Startup is generally estimated at 2 to 2.7ish times that or 8000 to 11,000 watts. This can last from several seconds to a good 1/2 minute. Fuses, generators and wiring don’t like that. Yours is likely on the higher side of which is just beyond the generator start up capacity likely about 10,400watts peak.
The secondary button must be a soft start to spool up the motor without high in rush currents. Sorry can’t guess at those numbers (don’t know enough about this)
Just remember you are running full load at 34amps at 115VAC and 17amps at 230VAC. More importantly without the button at start up likely 97 amps or 48amps respectively.
Reason for such high currents, this is a single phase motor, farm duty at that (meant for heavy work and higher stall loads). This was likely an aftermarket retrofit to meet local requirements.
Please ensure your feed wires are sized correctly for your safety, wouldn’t be good if your season is cut short because your rebuilding the house because of an electrical fire.
Disclaimer, no I’m not an electrician, however 10 years in the elevator business taught me a lot about motors and power requirements.