Yes, it does have the heat shield (but is the OEM one enough)?
Granted, I am a novice, but have been mucking about the internet researching similar symptoms and it all seems to be explained by the heat generated from the block evaporating the fuel in the bowls. Somehow I can't accept this, as I can stick my hand between the block and the heat shield without getting burned, and yet the car will not start again until it is cooled down, or wose yet, push started (if we are at an intersection blocking traffic).
Carb tuning? Distributor? Anything? I am up to documenting and taking pictures of any procedure you suggest and posting the successful one on a website for future refence to help out future MGB'ers with the same problem.
Or maybe you it's just as "simple" as going and getting my carbs professionally tuned, I dunno.