You either pay it or you fight it. Remember, the Keeper (your partner) has received a speculative invoice from an unregulated private parking company. There is no legal obligation on the Keeper to name the driver to a private company. They are not an authority and cannot issue "fines".
I can tell you from many years experience of doing this that UKPC will issue a claim if POPLA don't accept the appeal that UKPC will reject. However, they will use DCB Legal, bulk litigator, who will eventually discontinue the claim as long as it is defended.
They hope that you are low-hanging fruit on the gullible tree and will capitulate and pay them once they initiate litigation. As long as the claim is defended, they will discontinue before that have to pay the trial fee. I say that with greater than 99% confidence.
What you have shown us is not the appeal rejection but an attempt to scam you into revealing the drivers identity. Unfortunately, you have not shown us the original Notice to Keeper (NtK) or what you put in your appeal.
In the meantime, you should reply to that request and tell them to go sit on a sharp stick and rotate. Or simply ignore it and wait for the appeal rejection and a POPLA code. Your choice.
If you search the forum, there are plenty of example of
Arkell v Pressdram (1971) style responses you can send to UKPC.