There can never, ever be a "magistrate" involved in a civil contractual dispute. Magistrates only ever deal with criminal matters in the magistrates court. Civil contractual matters are dealt with in the county court by a district judge. Very big difference and the two should never be confused.
That aside, as already advised, you should go with Plan A for now and get the management at the Lidl store to get this cancelled. They have contracted ParkingEye and you should complain that as a customer on the day, irrespective of your mitigation, which, to be frank, no one really cares about and certainly ParkingEye don't. You may not want to mention this to the Lidl management either as they can simply say they don't care either and therefore it is your fault for overstaying.
Don't let some underling customer service bod fob you off and tell you to simply appeal to ParkingEye. Unless you get a positive result from the management at the store, you escalate your complaint as high up the management food chain as you can go and so you email the CEO.
At the same time you can appeal to ParkingEye, without the spiel about the doctors phone call and simply say that it is unfair and you were a paying customer of Lidl and show them evidence of your receipt. Nothing else.
If they don't cancel, you will get a POPLA code where you can then make a secondary appeal but at that stage, it 50:50 whether it succeeds. There are issues with all ParkingEye Notices to Keeper (NtK) that make them non PoFA compliant and also their signage is not PPSCoP or BPA CoP compliant. All arguments to make if Lidl or ParkingEye themselves don't cancel it.
If it gets to that stage, it will most likely end up as a claim which will be easily defeated because ParkingEye will not use their in house legal team to submit a claim and they will instead use DCB Legal which is a guaranteed with a reasonable defence and they eventually discontinue.
So, Plan A... complain as high up the management food chain at Lidl and get them to call ParkingEye off. Plan B, you appeal with the following just before the appeal deadline to ParkingEye themselves but only as the Keeper, referring to the driver in the third person only:
Dear Sir/Madam,
I am writing as the registered keeper of the vehicle regarding Parking Charge Notice [PCN Number] issued for an alleged overstay at [Lidl Car Park Location] on [Date].
The driver of the vehicle was a genuine customer of Lidl and has retained the receipt as proof of their transaction. It is unreasonable to issue a parking charge to a legitimate customer simply for taking the time necessary to complete their shopping. Lidl provides parking for its patrons, and penalising them in this manner is unfair.
Please find attached a copy of the receipt as evidence that the driver was a genuine Lidl customer on the date in question.
I request that this charge be cancelled. Should this appeal be rejected, I will escalate a formal complaint to Lidl regarding the unfair treatment of their customers.
I look forward to your confirmation of cancellation.
Yours sincerely,
[Your Name]
Registered Keeper
Poor signage, PoFA non compliance etc. can all be dealt with later if Lidl or ParkingEye don't cancel it first.