Your generator needs 12v to power its fuel pump and thus continue to get fuel. When the RV engine is running, the alternator provides 12v. When the engine is off, the generator is relying on one of the batteries to provide 12v to run the fuel pump. My guess is that battery is low and runs down after 15 minutes. My generator is powered from the chassis battery, but f you can restart the engine the chassis battery must be ok. Maybe a house battery? Or a separate battery just for the generator?
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL