Well, I wanted to follow up on this... The mystery is solved, finally.
After doing a BUNCH of things without 100% success, the one that finally worked was replacing the fuel pump. When I took the old pump apart, I found that the linkage inside it was loose in such a fashion that it could rotate, and if it rotated just right it lined up with a keyhole in the arm that's activated by the rocker, at which point it would cease pumping.
Apparently, given enough tinkering and swearing, it would once again rotate to where it would work, and I would think it was fixed for a period of time.
Since replacing the pump early this summer, it has worked fine.
If you look at the second image, you can see the "T" coming through the arm. First image, not there because it's rotated and went through the hole.