What actual evidence have they provided to prove that the "driver" left the premises?
All that effort you put into wasted appeals that were never going to work! The burden of proof is on them to prove the breach of any term, not for you to disprove. What a waste of effort.
You haven't even shown us the Notice to Keeper (NtK) that was sent. That will have the wording that they are relying on in their allegation.
Their sign states "Customer use only while on the premises. Anyone seen to be parking and walking off site will be in breach of the terms and conditions". How can that even be an enforceable term in a contract?
Ask yourself, if the driver parks but only a passenger walks off site (not that any boundaries are defined as to where the actual "site" is delimited), who is the person deemed to have breached the terms? It’s unclear who exactly would be breaching the terms if only a passenger (and not the driver) leaves the premises.
A contract cannot reasonably hold the driver liable for the actions of other occupants unless it explicitly specifies that responsibility. Even if it did, such a term would be open to challenge for being unfair, as the driver cannot control the actions of passengers in all circumstances.
Without clear delineation of the "site" boundary, the driver (or passengers) have no way of understanding what constitutes "walking off site." A contract term that is ambiguous to this degree is considered unenforceable under the Consumer Rights Act 2015, as it lacks the necessary transparency for the driver to make an informed decision about parking.
Hopefully, you now realise that you are dealing with a scammer. This willingly be resolved in the ultimate dispute resolution arena, the small claims track of the county court.
Ignore any and all debt collector letters. They are powerless to do anything. Use them as kindling or line the bottom of a litter tray.
If/when you receive a Letter of Claim (LoC) giving you 30 days to pay or they will issue a claim, come back and we'll advise on the next steps. For now, get on with your life and don't worry about this. It is easily defended when the time comes.