Would guess it was only intended to be a quick pick up (ended up being a 15 min stay). The street in question is narrow and a dead end so opportunity to turn can be tricky hence using the in and out of the car park.
But why did it take 15 minutes to pop in and out of the car park if it only did so intending to “turn” due to the narrow confines of the dead end street?
Personally, I am loath to pay a penny to the scamming ex-clampers in most situations. However, you have not provided enough evidence of their behaviour being outside the scope of their remit.
The NtK appears to be compliant with the requirements of PoFA to be able to hold the keeper liable if the driver remains unidentified. The 15 minute stay has not been explained other than as a parking event. You have not explained any other extenuating circumstances that can be explored.
You have to decide whether you think it is worth the £40
bribe discount or that the FCN is is so unjust that it is worth fighting it all the way up to a county court claim. It may never get that far.
The PPC could decide that it is worth threatening the keeper with DRAs or even a claim in the hope that it will be enough to scare the keeper into paying the, by then, inflated charge, but may then just give up as not worth it to pursue further. Who knows?
It may be worth seeing what other regulars think.