One other feature to check for on these cars is the Gearbox Case. Apart from being a 5 speed, the cases were uniquely cast and should have the following features.
The Casting No. on the casing, while being a DAM5626, also includes JKD initials

also, on the bottom, there's 2 bolt heads

At least 1/2 the Minis are unique, because owners do things to them.
Rare minis are early ones.
Ha ha. Oh the negativity.
It inspired me to try and fit 5 gears into a Mini box.
My good friend Dave Rosenthal did 6 - with a sequential shifter - does that count?

Yes, I went down the motorcycle route and got as far as ascertaining that the cogs would fit
The first problem is that they don't have a reverse gear.
The second is that most motor cycle gear boxes drive in and out at the same end. but the BMW K has the input and output at either end, so that seemed a good start.
It uses a 3 shaft system, so the input could be extended to a reverse gear cluster, and a bonus was that I was told, erroneously, that 5th was a 1.6666 overdrive.
It turned out that 5th is a reduced 1.6666, which means that the transfer gears have to be used to bring it back to 1:1, and that entails chopping and changing the transfer case.
I thought that my "prototype" had been destroyed in a barn fire, but about a month ago when rummaging in some boxes, I found it.