I have a sneaking suspicion (no more) that the restriction on learning on a Private Cat or Permit aircraft is to do with the payment of the instructor. If the instructor isn't paid for his efforts, then there may be a way.
You'd need to check the rules on this, but there may be a chink there.
The other suggestion is: talk to the CAA. You're not going to do anything illegal anyway, so have nothing to hide, and may find someone with the knowledge, authority, and inclination to be helpful.
It's a wonderful idea, and I hope it takes wing!