Shadow II unusual fuel leak on RH carb.

I'm wondering if the additional weight of gas is causing the float needle valve to open by causing too much fuel pressure - just a wild guess. Interesting problem.

P.S. Really like your respray - thanks for putting up the pics and details.

the needle valve in the carb will not close strong enough.

The needle valves in both carbs are probably worn, and the one that's leaking has just gone over the edge.
While you're in there, check both floats, and make sure there's no gas in them (leaky holes).
There's a brass screen that may need cleaning too.

Get the parts from Burlen in the UK and make sure that the float is set to specification.

I'm thinking that the float chambers and all the moving parts have not been looked at in decades.