Here's what i would do, Locate the ignition coil, it should just be removing the side blower housing cover. Trace the spark plug wire if you need help locating it... there should be 2 wires on the coil. The spark plug wire and the coil Kill wire.
unplug the coil kill wire. and start the engine. If it dies, you then know it is a problem fuel problem, or fuel solenoid problem. If it Does not die, there is something grounding coil, such as a bad low oil switch, or may even be the coil is failing when it heats up.
Can you start the engine immediately after it dies?
Yes it will start back up on the first pull. It is model Powermate PM0125500
I'll give what you said a try.