Thanks for your comments on this, Timmy. I've learned more than I expected! I rang Mini Sport and they told me there was no long neck cap for overflow modifications.

The original style mini cap (like the one in the photo you first posted) does have a small spring-loaded valve in the centre of the main sealing ring assembly. I assume it is meant to allow air to pass into the radiator when the pressure/temperature starts to drop - in the same way that the recovery caps also have that valve to allow coolant to move to the radiator.
The real difference (I'm making this up now) is that the modern recovery caps have a seal under the top of the cap to prevent air entering the filler neck and maintaining a vacuum to draw liquid back in from the overflow tank.