It sounds like a cooling problem. I would run the gen stopped. Then check the cooling air exaust vent(s) to be sure there is a GOOD air flow and there is no debris blocking those vents. If you can get to the intake, check there also for any restriction.
When driving on a hot day, the air off the road is going to be hotter, so even a small cooling restriction could trgger a hot engine shut down.
John & Clare - Buck & Zeus - Jack & Husky
2008 Fleetwood Bounder 38P
Towd: 2 HD Streetglides