This was what the "breach notice" wrote:
"As we have previously explained, The breach notice fee is the cost of enforcing the car park's terms and conditions of parking. This fee is payable under the contract and is not a penalty, a fine, nor a claim for liquidated damages."
So they aren't claiming for liquidated damages, presumably because they can't demonstrate that. If it comes before the DT, one of the points I will argue is for them to demonstrate how the charge relates to the cost of enforcing the car park, when 100% compliance would mean that they are losing money to pay for enforcement. Logically, the costs of enforcement must be built into their overall operating model. The standard parking charges pay for their operation, which includes enforcement as that is central to their operation.
IANAL but one thing I wonder is, if there were available Car Parks for others to park in, how have they lost anything. Or would that fall under liquidated damages?
I'd agree that your argument could be that the costs of running their business is built into costs charged for the services. If they didn't get any breach fines on a given week due to no-one overstaying, they cannot claim they had no revenue to pay their enforcement staff.