Bryan,
Just a thought...when you rebuilt the carb, were there any internal dimensions for the accelerator pump stroke/hieght? It's been a while since I looked at the specs, but seems like there is an aadjustment on this and it is often overlooked. Timing is critical in the pump shot/stroke.
Also, these old carbs are worn quite a bit, especially in all the linkage holes. Could it be that when you push the pedal, the first movement (when nothing happens) is just all the slack coming out of all the linkage connections?
Just food for thought. Let us know how it turns out when you solve it. Jim