You didn't mention what year the camper is.
In my 1991, the fuel line for the generator is tee'd off the fuel line coming out of the front tank (which means it CAN run the front tank dry). I'm not sure how it interacts with the tank's fuel pump (i.e., whether that pump runs or if the generator's fuel pump just pulls it through). The generator has an internal fuel pump, as I understand it.
I don't know how later years with only one tank are hooked up.
1995 Airstream Classic 30' Excella 1000
2014 Ram 2500 Crew Cab with Cummins 6.7L Diesel
Sold but not forgotten: 1991 Airstream B190
Sold: 2006 F-250 6.0L Powerstroke Supercab