A good first step, 'Plan A' as it's often referred to, is to speak to the manager of the Waitrose store, taking along a receipt from the shopping done in their store on the date in question, and asking if they will intervene. Be polite but persistent.
If you do not get anywhere with Waitrose, then make sure you don't miss Britannia's deadline to appeal. They do not seem to be seeking to use Schedule 4 of the Protection of Freedoms Act (PoFA, there's a link in my signature at the bottom of this post) to hold you liable as the registered keeper, so you could appeal as the keeper along the lines of the below:
Dear Sirs,
I have received your Parking Charge Notice (Ref: ________) for vehicle registration mark ____ ___, in which you allege that the driver is liable to pay 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 the charge has been cancelled. 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 28 days, chase them.