The fact that there is no "engine number" (we are actually referring to the duplicated frame number, as the frame number is master, and the so called engine number is just a copy of the frame number) probably means that this was a spare engine (so it wasn't fitted to a frame yet, hence no duplicated frame number)
The BB marking is still defeating us. Some engines have no B marking at all, some have a single B, some have a double B. We haven't found any logic yet. It could be a BSA inspector's stamp...?