There is one point, which I will come to. However, I think you'll be hard pressed to say that there is insufficient notice that customers who wish to stay longer than 90 minutes need to input their VRM into a terminal within Costa.
For your Plan A, you should really be aiming as high up the Costa management food chain as possible. Find an email address for the CEO and start there.
Regarding an appeal point that may or may not work for Plans B and C is a failure to fully comply with the strict requirements of PoFA 9(2)(e)(i) as I cannot see anywhere in that NtK an invitation to the keeper to pay the charge. Because of this, they cannot rely on the provisions of PoFA to hold the keeper liable. As they don't know the identity of the driver (who is liable) they have nowhere to go.
Another point will depend on where exactly this Costa is actually located. If it is indeed in the car park of the airport, it may not be on "relevant land" for the purposes of keeper liability.
So, under no circumstances identify who was driving. Any appeal or communication is from the keeper only.
Having just tried to find the exact location of this Costa Coffee based on the location identified on the NtK, I am confused. Where exactly is this Costa coffee? A Google Maps search reveals only a Costa within the terminal. The NtK says "Costa Coffee Southend Airport, Southend" and on the back of the NtK it states "Costa Coffee Southend Airport, Southend Car Park".
With a location that is undoubtedly ambiguous, you have another PoFA appeal point (9(8)(a)) that should get this cancelled at Plan C, POPLA stage. I had one like this recently and appealed to POPLA where ParkingEye withdrew the PCN before adjudication. If it comes to that, I will provide you with the wording of that POPLA appeal.
I must stress that any PoFA appeal points are only valid as long as the drivers identity is not revealed, inadvertently or otherwise.