Your dealer is full of crap. I have seen gensets running full load at 120. No problem. You have another problem.If your generator fuel line is teed into the main engine supply line. The main engine could be draining the generator line.
There is no fuel bypass system on your unit. From pump to carb. Fuel flow controlled by needle and seat in carb. Trash under seat is a common problem. As you have found out. Cleaning the trash out will solve your problem.
I agree check the oil level. Onan had that problem years ago on one of there units. Oil pan was not drilled so oil could flow back to the pickup for the pump. Ran fine sitting still. But make a turn and it would shut off because of low oil. You most likely have low oil.
Your part will have the number on the back. It is the black box under the start solenoid. Check with flight systems they may have a cheaper price. Also they have a better warranty. Before you change it out. I would look for bad connections.