I'm not familiar with the sidedraft Webers used on the spitfire, but I am quite familiar with the (downdraft) Weber IDFs that are used on VWs and Porsche 912s. So, this may or may not be relevant, but since no one else has answered, I'll give it a shot.
Since it starts and runs at idle, it's not ignition. Since the car dies when you open the throttle, it seems that it has to be a carburetor problem. I would look at the various jets to make sure nothing is plugged; the jets on Webers get plugged very easily. Do you have a fuel filter on the fuel line from the pump to the carbs?
I'd look especially closely at the accelerator pump jets. If one or both of those are plugged, the mixture leans out when you open the throttle and it can stall the engine. It's possible that the idle and progression jets are plugged, too, even though you are getting enough fuel through them to keep the engine running at idle.