When it dies out I would do spark and compression check. IMO blowers require alot of torque and power to keep the turbine spinning at high speed. Other than carb. adjustment after it warms up would be the only suggestion I have if it's got good spark and compression when it acts up?

Assumming the carb is rebuilt correctly.

1) Take off the muffler to check the exhaust port for buildups.

2) Close the choke half way and rev it and see whether it help. If so, it is too lean or air leak.

3) Get a can of Deep Creep ( spray Sea Foam), run the engine until it is warm, spray along the seam between the cylinder and the crankcase and the lower crankcase. I think you have the same engine as the HomeScaper trimmer engine. I have one on hand. It has separate cylinder and upper and lower crankcase. Spray along the gasket to observe any bubble while running and reving to check for air leak.

4) Tighter the two screws of the carb to make sure it is tight.

5) If nothing help, check spark.

6) If everything seem fine. Then go back to the carb!!! You're screwed!!!!