If the carcass is being donated in-skin, then the exemption probably applies. If however, you'll be skinning it, then you'll need to be registered with your local council as a Food Business I believe.
The fact it's being donated, not sold, makes no difference - it's still supply to a third party.
(The difference between skinned and unskinned is that, on the one hand, you're supplying wild game, and on the other hand your supplying wild game meat.
The extract of the rules, posted by
@Stalker1962 in post #5 is the bit that refers to wild game, not the bit that refers to wild game meat).