Looking at those terms, it doesn't say that parking is free for blue badge holders. It says that there are reduced rates and you need to apply for them. Did you or your passenger apply?
Have you tried Plan A, which is to contact the venue and ask them to cancel the PCN? As far as I can see, that is going to be your best option.
Whilst displaying a blue badge indicates that there is a person with either reduced mobility or a protected characteristic using the vehicle, on private land it has no other value. On private land, it may be necessary to display the blue badge if you intend to use a disabled parking space but, aside from that, it does not confer any rights that may apply when using it on land under statutory control.
The NtK is PoFA compliant and so liability for it can be transferred from the driver to the keeper. I doubt you'd have any success appealing it, either at Plan B stage (with the operator_ or Plan C stage (with the IAS). Whether Gemini would progress this as far as a court claim (Plan D), I don't know. As far as I'm aware, Gemini are not particularly litigious, at least for a single PCN.
Try Plan A first. Explain the circumstances and explain how it may be confusing to understand the requirements for blue badge holders. stanborough@gll.org