Is the generator level? Unlevel may cause low readings to the fuel level and/or oil level sensors, forcing shutdown.
Here are some of the possibilities from the troubleshooting section for Onan BGM and NHM sets. Not knowing if yours is either one, I'd still think it's all pretty generic in the control section, maybe it will give you somewhere to start. This may be stuff you already checked.
Trouble: Engine starts and run, then stops. Set re-starts immediately, or after cooling down.
Possible Causes:
1. Fuel level or oil level is low.
2. Dirty fuel filter restricting fuel flow.
3. Faulty ignition module
4. Contaminated fuel
Corrective Action:
1. Check fuel and oil levels, refill if necessary
2. Clean or replace fuel filter.
3. Replace ignition module
4. Refill tank with fresh fuel.
Good luck with it!