If the bulb makes no difference, but the problem is fuel starvation, that would mean there's no fuel in the bulb when the problem occurs.
1. Does the bulb squirt fuel when the engine is not running? (I think you can see the fuel squirting by looking in the open end of the carburetor, where the air cleaner would go).
2. If you squirt a little fuel into the carburetor air opening when the engine is running and is in the process of quitting, does it enable the engine to rev up briefly? Naturally you need to do this test carefully, not getting any part of yourself in front of the open end of the carburetor, and using an oil can or something similar to control the fuel delivery safely. If you don't feel comfortable you can do this test safely, don't do it.
