I've split this off into its own thread in line with #2 of the House Rules (https://www.ftla.uk/announcements/house-rules/):
2. Do not hijack another person's thread in order to post your own problem. Start your own.
You may also wish to edit your post in line with the advice in the private parking forum guide - READ THIS FIRST - Private Parking Charges Forum guide (https://www.ftla.uk/private-parking-tickets/read-this-first-private-parking-charges-forum-guide/)
In terms of the alleged breach, it would seem the driver is bang to rights (without seeing the signage - if you have photos these would be useful to see). You could try speaking to the management at the store - they might not be overly keen to help, but equally there's little to lose by trying.
I assume you are the registered keeper of the vehicle? If so, there is a technical argument open to you. Excel will probably reject it - but the IAS may well uphold it (although they are often described as a kangaroo court, so there remains a chance they'll reject it too - make sure you're aware of this when choosing how to proceed, the decision is yours to make). If you wanted to go down this route, you could send something along the lines of the below, as the registered keeper:
Dear Sirs,
I have received your Parking Charge Notice (Ref: ______) for vehicle registration mark _______, in which you allege that the driver has incurred a parking charge. I note from your correspondence that you claim that you will pursue me on the assumption that I was the driver, if you do not know the name and address of the driver. There is no requirement to name the driver, and I will not be doing so. In addition, no reasonable assumption can be drawn as to the identity of the driver.
If you wish to pursue me as the registered keeper, you should do so under Schedule 4 of the Protection of Freedoms Act. The notice you have issued does not comply with the requirements of Schedule 4 of The Protection Of Freedoms Act 2012 (PoFA) namely, but not limited to, failing to deliver the notice within the relevant period of 14 days as prescribed by section 9 (4) of the Act.
I am therefore unable to help you further with this matter, and look forward to your confirmation that the charge has been cancelled.
Yours,
There's a link to Schedule 4 of PoFA in my signature at the bottom of the post, which is worth reading to understand the argument above.