Re: engine cuts out under load
It is normal for the bulb too lose "stiffness" when the motor is running. The motor sucks the fuel through the bulb and so there is no pressure in the line to cause the bulb to be firm. The bulb, however, should not collapse when the motor is running. If it collapses, that is a sign of a fuel line restriction upstream of the bulb.
If the motor does not respond to squeezing the bulb when the motor cuts out, then I would guess that it is not a fuel pump problem and more likely an ignition problem, link and sync problem, or a gummed up carb.
Just some thoughts...