Where are the "signs" they mention in the NtK? "The signage which is clearly displayed throughout the area..." Errr... not from the photos you've shown us, assuming they are contemporaneous.
The contract can only be formed if the signs are prominent and adequately notify the driver of the charge for breaching the contract. Is there any sign when you enter the location that clearly informs the driver that they are entering private land and that parking is controlled by the operator and tells the driver to see the signs for the terms and conditions of parking?
What was the "period of parking" immediately preceding the incident time? Was that period one minute, one hour or even one year? What is the definition of a "period"? "A length of time". As no "length" of time has been defined, no "period" has been defined.
So, for now, there are two clear points to appeal on, but only as the Keeper. Under no circumstances appeal as the driver. They have no idea who the driver is unless the Keeper blabs it to them. They can only hold the Keeper liable if they fully comply with all the requirements of PoFA. As they have not defined the "period of time", the NtK is not fully compliant. Partial or even substantial compliance is not sufficient.
Also, no contract can have been formed if the signage is inadequate. If there is no contract, then there is no debt.
Unfortunately for you, UKCPM are IPC members and nether of them are going to accept any appeal. You can try but if they refuse, you simply have to wait and see if/when they try a count court claim. That is a good thing because the county court is the ultimate dispute resolution service and not something to fear or worry about.