First, can we clarify - you mention a POPLA appeal. In order to appeal to POPLA, you first must appeal to ParkingEye. Have you done this? If so, what exactly did you say in this appeal?
POPLA won't entertain the 8 minutes argument, but if it can be made credibly, a court may well do. Mobile payment apps/websites can be horrendously clunky, even with a good signal, so entering the car park, finding a space, reading the entirety of the terms and conditions, attempting to pay, failing to do so, returning to the car and then driving out could quite easily take 8 minutes, in my view.
Speaking of the terms and conditions, do you have any photos of the signage?