Normally I would agree with you on this point... but, as I understand the scenario given by the OP, parking is part and parcel of his overnight stay at the hotel. His car therefore has not been parked illegally. The customer cannot be held accountable for a sub contractors inept computer system
If the OP has done everything asked of him, and can prove it, then it is upto the Parking gimps to take him to court to answer their evidence (or lack thereof). When the case goes against Parking Eye, then any time and trouble, plus damages will be awarded to the OP.
I would also suggest, that as the appeals have been upheld, then each case will go no further anyways. Keeping records is a protection against incompetence, rather than a defence of wrong doing. Should the need to appeal via POPLA, then historical proof of Parking Eyes ineptitude can only strengthen any appeal against them.