Is there any chance you can get some photos of the location where you parked? If it was an accessible bay, it would be worthwhile getting some pictures of any signs there. There has to be a sign with the terms and conditions that is readable from within the vehicle. Many operators do not comply with this requirement.
However, the PN is not about the blue badge. It is because the vehicle was parked fro longer than the permitted period free parking. The fact that the driver required more time due to a protected characteristic, is a good appeal point.
Also, the Notice to Keeper (NtK) is not fully compliant with all the requirements of PoFA to be able to hold the Keeper liable and they have no idea who the driver is, unless you, the Keeper, tells them. As there is no legal obligation on the Keeper to identify the driver, you should not do so. Only refer to the driver in the third person. No "I did this or that", only "the driver did this or that".
As any initial appeal will be rejected, no matter what the circumstances, it is not worth wasting much effort on that. However, the rejection must contain a POPLA code which will be valid for 33 days from the date of the appeal rejection.
For now, I suggest you appeal along the following lines:
There is no legal obligation on the
known keeper (the recipient of the Notice to Keeper (NtK)) to reveal the identity of the
unknown driver and no inference or assumptions can be made.
The NtK is not compliant with all the requirements of PoFA which means that if the
unknown driver is not identified, they cannot transfer liability for the charge from the
unknown driver to the
known keeper.
Use the following as your appeal. No need to embellish or remove anything from it:
I am the keeper of the vehicle and I dispute your 'parking charge'. I deny any liability or contractual agreement and I will be making a complaint about your predatory conduct to your client landowner.
As your Notice to Keeper (NtK) does not fully comply with ALL the requirements of PoFA 2012, you are unable to hold the keeper of the vehicle liable for the charge. Partial or even substantial compliance is not sufficient. There will be no admission as to who was driving and no inference or assumptions can be drawn. UKPA has relied on contract law allegations of breach against the driver only.
The registered keeper cannot be presumed or inferred to have been the driver, nor pursued under some twisted interpretation of the law of agency. Your NtK can only hold the driver liable. UKPA have no hope at POPLA, so you are urged to save us both a complete waste of time and cancel the PCN.
When the appeal rejection comes through, you will be able to appeal to POPLA where a fuller argument can be made.