I'm not really sure if this question is inappropriate, but does anyone know if there is a way to harden the pinky-side edge of the hands instead of the knuckles ?

Any time someone wants to harden the skin they use the same techniques.

It is not where you want to harden, it is that you want to harden.

The problem with such training is, it can leave you with nerve damage.

Sure you don't feel blows, but perhaps
You would better off spending time learning technique.

There exist certain techniques to not allow as much of the force of the impact to remain in you,
Therefore transmitting more to the target, so being more effective for the same amount of effort.

You want to be more effective, not more restricted, so my advice is to not work on hardening the hands.
Work on technique. You can learn when to flex and when to relax muscles and thereby do well.