I think you may have confused keepers and drivers somewhat in your post - it would be the driver who is required to enter the VRM into a terminal at the car park. The only role the keeper plays is they're the person who receives the notice.
'Plan A' is generally to try and speak to the management of Costa to see if they can intervene - ideally do this without revealling who was driving, and providing a receipt to show the driver was a customer as was not simply misusing the car park.
That notice is rather late - can you confirm that you are the vehicle's registered keeper, and that you have the vehicle's V5C document in your possession with the correct address? If so, an appeal as the keeper along the lines of the below should be a good idea:
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 are not seeking to hold me liable as the registered keeper, under Schedule 4 of the Protection of Freedoms Act 2012 ("The Act"). You have chosen not to issue a Notice to Keeper in accordance with The Act, and it is now too late for you to do so.
There is no obligation for me to name the driver and I will not be doing so. I am therefore unable to help you further with this matter, and look forward to your confirmation that no further action will be taken against me. If you choose to decline this appeal, you must issue a POPLA code.
Yours,
If appealing online, be careful there are no drop down/tick boxes that cause you to identify who was driving, and keep a close eye on your spam folder for their response. If they do not respond within 35 days, chase them.
