I presume you won't be patronising the establishment for the wedding venue.
It is good that you appealed only as the keeper. As long has you have not identified the driver, this will be won at POPLA.
You have to be careful when reading the response to the appeal:
...you have stated that you had registered your vehicles details on site on the date of the above contravention.
...between yourself and the parking site, with each visit that you make and complete, entering and exiting.
Notice that they try to implicate that
you were the driver, which, unless you've inadvertently admitted to being, they have absolutely no idea.
In your POPLA appeal, you only need to show that the Notice to Keeper (NtK) does not comply with all the requirements of PoFA 2012 and therefore the keeper cannot be liable for the charge. As you are under no legal obligation to provide the drivers identity to an unregulated private parking company, the burden of proof is on the operator to show that the person they are pursuing is the driver and no inference or assumptions can be made.