Have you checked their website for evidential photos? Go as though to appeal (but don’t do so yet) and if they have any evidential photos, they must be available there.
As the Notice to Keeper is not fully compliant with all the requirements of PoFA, they cannot actually hold the Keeper liable. The only way they’d know the drivers identity is if the Keeper blabs it to them, inadvertently or otherwise. The Keepr must only ever refer to the driver in the third person. No “I parked here or there”, only “the driver parked here or there”.
Did the driver or a passenger have a blue badge or a protected characteristic as defined under the Equality Act?