It does sound like heat is the issue, vapour lock?
The weber carb kit comes with an inline fuel filter which sits directly over the manifold. If you have one I'd remove it even just fir safety reasons.
1. Check fuel tank isn't rotting from inside out, big chunks can clog up the carb. Mine would do exactly as you've described. When I turned it off, the fuel would return drawing the lumps of crust away from the carb and it would then run well for a bit.
2. Clean the carb
3. Make sure there's no air leak around the carb.
4. The heat shield around the manifold was missing on mine, I think it helps direct heat away from the carb.
5. As suggested replace the ignition coil, leads and plugs, points.
Let us know how you get on.