BSA owners club tech section, says swing arm is from a later bike with full width hub. Didn’t say which bike or year, not so helpful. Looks like the easiest option is to make a top hat bush for the spindle to frame lug to eliminate the space and probable make a spacer from wheel to lug same dimension as a plunger spacer but with 7/8 ID.
Anyone know the length of that spacer, bets are on 1.403 or there about
Phil