Did you check if the plating was nickel or chrome? Chrome would be a 1930s or later bike usually, nickel an earlier bike. Did it have the white lower portion on the rear mudguard? The BSA catalog for 1934 shows black rear mudguard with a different rear reflector, but the 1936-39 catalogs show white rear tips and a reflector more similar to yours.
Frame serial numbers are notoriously hard to use on British bikes, more so than American ones. We usually date based on hub/component date stamps, component styles, paint schemes, decals, and type of plating. But it sounds like you have at least some idea from the frame date, which is better than most.
That looks most similar to the bikes in the late 1930s-period catalogs, particularly 1936-39, including the two-tone silver head tube. My guess would be a late 1930s bike. That sounds consistent with the bit of information you have from the serial number.
I'd have a hard time selling that one. We're hard-pressed to find that kind of cycle here in the U.S.