I don't think BSA fitted anything like a conventional breather on these engines. The left side rear cylinder base stud is hollow and contains a ball bearing which acts as a valve. As the piston descends the bearing lifts off its seat and the oily breath passes up the pipe to lubricate the inlet valve stem. If the ball was stuck open, could you be getting excess oil delivered to the cylinder? BSA might have relied on the  crankcase volume increase provided by the wet sump to "absorb" pressure. Otherwise, some of the crankcase pressure might pass through the drive side main bearings into the primary case, exiting via the chain oiler pipe in the rear of the primary case. I had the same concerns about engine breathing, so made up a ball valve breather fitted to the  timing side inspection cover, with a pipe leading to the exposed exhaust valve. Not original, but it works! You could maybe use a Royal Enfield breather duckbill valve. Good luck!