Continue to only appeal as the Registered Keeper (RK). They have no idea who the driver is and you are under no legal obligation to identify the driver to an unregulated private parking company.
If we put it together, PoFA paragraph 9(2)(a) states:
9 (1) A notice which is to be relied on as a notice to keeper for the purposes of paragraph 6(1)(b) is given in accordance with this paragraph if the following requirements are met.
(2) The notice must—
(a) specify the vehicle, the relevant land on which it was parked and the period of parking to which the notice relates;
So, is there anything in the NtK you received that "specifies" the "period of parking"? You tell me.
Unless ALL the requirements of PoFA are fully complied with, then they cannot hold the Keeper liable. So, don't identify the driver.