I have measured the clearance between the inside of the frame rear wheel lug and the backplate on my 1930 2 Speeder and it is 0.450 inch. There is room for a 3/8″ nut but it must be positioned before screwing in the rear stand bolt if the rear wheel is in place.
My rear brake backplate is a club spares casting that I machined and it is a little thicker than the original one.
Coincidently I lost a rear stand bolt last September and the locking nut is essential if one is not going to loose the bolt. As it is a locking nut one can use a thinner nut if necessary, 1/4″ thick rather than the normal thickness.
I hope this helps.