I too have the very same identical problem with my '69 383 Bee. The engine was rebuilt 3 years ago. It too gets so hot under the hood that I melted the styrofoam seal around the steering column. I read in
the April issue of "Hemmings Muscle Machines" ask Ray: I guy wrote in to Ray with the same problem I'm having. Only overheats at idle or when I stop. Ray said: If you can suck a paper towell to the grille at idle, the fan is more than sufficient. I beleive the water pump is turning too slow, or the pump you have is designed for high RPM, and the impeller works inefficiently at low speeds. Ray went on to say here is how you test it. Raise the engine speed to about 2000 RPM when the temperature spike occurs and hold it there for a minute or 2.
If the gauge starts to drop then there is most likely a flow problem, not an air movement issue.
I have'nt tried this yet but am going to shortly. Try this and let me and everyone here know if it works.

Jim from South St.Louis