I misread my frame number... It begins with "E" not "B".
I found that same link, Adrian, but it is incomplete and doesn't account for my numbers. I found another link online last night that has a better, more complete, BSA serial number listing. According to it my frame and engine both appear to be '57's. http://www.bsaownersclub.co.uk/Engine_Frame_Listing.html#1950
My numbers are slightly lower than what's listed, but I'm going to assume they're both from '57.