Any initial appeal will be rejected, irrespective of the circumstances. You are not dealing with a firm that has any customer service ethos. All they want is to extract as much money out of you as they can. These are ex-clamper thugs who will do anything they can to try and scam you.
There is no legal obligation on the Keeper to identify the driver to an unregulated private parking firm. Irrespective of who was driving, you only refer to the driver in the third person. No "I did this or that", only "the driver did this or that". However, in this case, as you were not the driver, it may be a moot point.
It's pretty clear that a valid blue badge was on display and so once the initial appeal is rejected, you can make a secondary appeal to POPLA, where they may even assign a real assessor to the case and find that the evidence is overwhelming and the PCN should be cancelled. In any case, as already pointed out earlier, there can be no Keeper liability if the notice does not fully comply with PoFA. In this case, paragraph 9(2)(a) has not been complied with and you can refer the POPLA assessor to the persuasive appellate case law in Brennan v Premier Parking Solutions (2023) where the judge dismissed the claim because a single timestamp on the NtK does not show a "period of parking" and was therefore not PoFA compliant and the Keeper could not be liable.