The Notice to Keeper (NtK) does not comply with PoFA 9(2)(e)(i) which mens that the Keeper cannot be liable for the charge, only the driver. However, that argument will not work at the preliminary appeal and is doubtful whether you could convince a POPLA assessor to understand the legal nuance of that point.
That does not mean that you should just pay it. What about Plan A? Have you contacted the gym and asked them why you should keep your membership if their agent can simply issue £100 invoices to their members. The whole point of them contracting Highview was, I presume, to prevent unauthorised drivers from using the car park. As a member with evidence that you were at the gym that day, should be enough for them to call ff their dog and get the PCn cancelled.