Wow, $5600 for a generator repair !!!!!!
I am having the same type of problem on my Generac 7.5 quiet.. In a way. Push the start button and she'll start right up and when you let up on the button, she dies. Start it and let up on the start switch, but hold the heater button down and she keeps running till you let up on the switch and she dies. Notice, I call the gennie a she, for the same obvious reason, I call the MH she. She is expensive and needs a lot of maintainence.
So, I get my Generac tech out and we go thru the whole diagnostic thing and it is determined that the alternator isn't putting out enuff volts to keep it running and is probably gonna take a new alternator, if one can be found.
2001 National Tradewinds 7390 LTC
2002 Subaru Legacy outback Limited stick