There may also be a ball bearing in the passage below the accelerator pump that is stuck. Also a passage could be blocked. I'm not familiar with the 4 barrel Carter, but I'd be carefully inspecting those passages and find a factory service manual that shows the passages and how the accelerator pump bowl is filled.
Love the car! I could make my favorite LS joke here, but I shant. 🙂 It sounds like the fuel filter, on the carb, might be in backwards. Don't ask me how I figured that one out, once, a long time ago. It could also be the float is set too low or possibly a needle and seat that is stuck. Mikes Carburetor is an incredible resource! As an alternative, if absolute originality isn't that important to you; a guy could switch out to an Edelbrock carb. They are a new evolution of the WCFB's younger brother the Carter AFB. The current Edelbrock and Carter built AFB have a dual bolt pattern, but you might want to make sure the butterflies are in the correct positions when bolting to an original intake manifold.