Regarding the fact it should have a B plate, back in the 1960s the local councils were a lot less strict than the DVLA is now. It has been reported that many councils used up left over 3 letter 3 number regs which obviously are non year specific and registered them on 63 and 64 cars, so this is what may have happened.
Mike Brewer's dream car Mini Cooper S is featured on the cover of the September 2019 Mini Magazine:-
https://www.facebook.com/minimagazine/
He believes this '64 Cooper S was re-shelled probably in the 1970's into an earlier '61/62 shell but is planning to rebuild it into a new Heritage Mk1 shell as a car for him to keep forever.
"I'm not one of those aficionados that want to get the exact matching code on everything! People out there obsess over this stuff. I just want a car that I can enjoy every day and drive every day so that's what I bought!"I noticed that the car has a "numbers first" registration which a lot of local councils used when they ran out of "letters first" plates in 1961/62. (By just reversing the ones they had already issued),
But, if the car is a 1964 it should have a seven digit plate with a "letters first" and a "B" suffix. Unless it took the registration of the 61/62 shell that replaced the original. Does that mean that if the shell is replaced again it will need registering again?
Also nowhere in the article does it mention the engine size. The S came out in 1963 with a 1071cc engine and was not changed until August 1964 when the 970cc and 1275cc came out, so could be any of these.