Whilst we could do with seeing the back of the Notice to Keeper (NtK), it is absolutely worth fighting all the way. There is no way any contract could be formed if they cannot evidence that the vehicle was parked for less than the 5 minutes consideration period.
Also, no contract can be formed if the signage is prohibitory. This is just a contractual issue. Unless the signs offer something in return (consideration), no contract exists.
Also, the wording that we can see on the front of the NtK, is not compliant with all the requirements of PoFA (which is why we need to see the back) and so they cannot transfer any liability for the charge from the unknown (to them) driver to the known Keeper.
If you follow the advice, you won't be paying a penny to these scammers. For now, show us the back of there NtK.