From everything I'm seeing here, it's a 100% genuine Mk.5. The "B" on the barrel wouldn't necessarily mean "BSA" in 1945. M47/C would be the military BSA code by then. ( I could certainly be wrong here and stand waiting to be corrected ) You may see the M47/C (BSA) stamped on one side of the flash hider but don't worry if it is. Many Fazakerley Mk.5's are fitted with them and it is quite normal. The rear sight has simply been changed for whatever reason and it happens to be a post war BSA manufactured sight. These "B" marked milled rear sights were only made by BSA for their non-government FTR program post war. The correct rear sights for these turn up often enough on E-Bay...just keep your eyes open. If you run into a brick wall, I have a few around somewhere.
that would be awesome and i'd be humble/grateful if you had any lying around! I would definitely have to send you a "you are amazing" cake! Do you know by chance what that marking is in the 3rd picture? on the right of the receiver?