There's a couple ways to narrow this down that I like.
The easiest once you understand what's going on, is to use the priming lever on the transfer pump to determine whether the fuel is pumping fuel, pumping air, pumped up and not pumping, or sucking a vacuum immediately after it dies. You can tell this by how the priming lever feels, I've written it out a few times and can go back and copy it here, but I don't think I explain it very well. You can figure it out just as easy by taking the lines off and using one finger to plug the outlet, then inlet, and feel the difference in the tension. Keep in mind the lever pulls the diaphragm back the same way the camshaft does to suck the fuel in, then the spring pushing the diaphragm causes the fuel pressure, unless the fuel pressure is high enough to keep the spring compressed, then the lever moves with no resistance.
The other way is to tee in an old fashioned fuel pressure/vacuum gauge wherever it's most convenient in the fuel line, if it pulls too much vacuum when it's dying then you look upstream, otherwise downstream.
If it's not blowing black smoke when it stalls, keep looking at the fuel supply until proven otherwise. Does this keep running at idle as long as you don't use it?