Is the fuel in the bowl going away when the engine is HOT and cools, or will it vanish if you just start up and shut it down right away?

What is probably happening, is the fuel, sitting in the HOT carburator bowl from the hot manifold, expands, and overflows into the intake manifold. Then, on cooling, the fuel leval in the bowl is now lower than with the hot engine running.
When you try to start from cool, the vacuum from the engine, or vibration from starting, causes the needle, possibly stuck from overflowing, to drop, allowing the pump to refill the bowl.

Try replacing the needle and seat, and checking the float leval and drop. Replace the float as well, if the needle contact area is dimpled.

That should fix it.

Gordon Raymond