That doesn't sound like a battery problem to me. If the battery was so low that the gen wouldn't run, then how does it start the motor after cooling down?
There have been a number of common problems with these generators. Most are just routine maintenance, others are not. Fuel filters get plugged up from junk in the tanks. Fuel pumps that go bad from the factory. Wiring that rubs or grounds on the throttle linkage, causing an under voltage situation that initiates auto shutdown. Bad fuel that gums up the carburetor after sitting for as little as a month. Alcohol fuel that is drying seals in the carbs. Even people mistakenly putting E85 fuel in the tank.
The problem you are describing sounds like the gen is having problems when it is loaded. If, as you said, the shop ran it unloaded, then shame on them for not doing some kind of load test.
Start at the beginning with the simplest problem, and working your way to the more difficult. (I would start with fresh fuel, then filter and work toward the electrical)
Just my 2 pennies worth.
PS. My routine is on the first weekend of every month, I start the generator, and plug in an industrial heat gun (2000 watt) in the garage, and an 1800 watt space heater (plugged into the bathroom) and run the generator for a minimum of 20 minutes, then shut off the heaters and let it cool down for 5 mins and shut it off. The load makes sure that the windings in the generator get good and warm, and burn out any moisture that might be in there, along with getting plenty of fresh fuel into the carburetor, and getting the motor good and hot.
2013 Voltage 3905