You may have hit the nail on the head, Andy. I pulled the fuel pump (boy did I like working on things better when the engine was out!) and the rocker arm was jammed too far down. With a little push it snapped back into place and everything worked again, including the priming lever. I took the cover off and inspected the innards. Everything looked in its place, and the pump worked on the bench just fine. When I put the pump back, I physically felt the cam inside the engine with my finger and made sure the rocker arm was above/in front of it. Now the priming lever works correctly, and I was able to prime the whole system by hand.
Also inspected the float bowl covers again just to be sure. The front valve seat wasn't as tight as it could have been, and the rear float lever wasn't exactly at 7/16", but I don't think either of those was far enough off to be causing the problem. Corrected them nonetheless.
Anyway, it's too late tonight to start her up again (The kid is asleep, and a three year old's sleep is first priority!) so I won't get to see if I've still got a problem until probably tomorrow afternoon.
I did get the Spitfire's carbs rebuild tonight too, so all I have left with that is to tune them and I should have at least one Triumph back on the road!