The problem is probably sonething as simple as an elctronic component thatis cracked or broken and the motor dies the second it heats up. The darned part is probably worth $1.
If you knew what it was and could get the part and put it on yourself the repair would be virtually nothing.
The problem I have with ALL new small engnes B&S, Honda, Kohler etc is that I don't have a clue about the electronics and even less idea of how to check each component other than swap them out.
Then come the kicker, the local guy charges $120 just to look at the thing. Then instead of removing said bad part he replaces a circuit board with eoth little electronic gizmos on it and when they are done it cost me $150 to get the darned motor running. Still it's cheaper than a new one.
Sort of like new cars. They run longer, get better milage and are easier all the way arounbd, UNLESS you have to or need to work on one.
There used to be a guy in Bruneau that did everybodies mechanical work in the valley. He finally retired simply because he couldn't keep up on the purchase of each and every individual tester or computer that was unique to each and every make of vehicle.
I think you motor proably has yerars left as long as you didn't use bad fuel and kept the air cleaner clean. The problem is going to be finding the ONE little diode or whatever that causes it to die after starting. :-(
Good Luck
RWT