The driver visited a hospital when they parked in a bay which turned out to be for Blue Badge holders only.
If an occupant of the vehicle was a patient, then speaking to the PALS team at said hospital would be a good first step.
Your references to Schedule 4 of the Protection of Freedoms Act and keeper liability in the appeal suggest a slight misunderstanding of how that process works. What has been received appears to be a windscreen ticket, aka a 'Notice to Driver' - they are not claiming the keeper is liable, as the PCN has been issued to the driver. Paragraph 9 to which you refer, specifies the requirements for a Notice to
Keeper, issued by post, so is not relevant here.
I imagine they have sent the response that they have on the basis that they have issued a charge to the driver, but instead have received a response from someone claiming* to be the registered keeper. *I say 'claiming' here not because I dispute that you are the keeper, but because at this stage, Gemini have no confirmation that you are the registered keeper, other than your say so.
You may wish to remind them that the BPA Code of Practice requires them to respond to appeals from drivers and keepers.
Have you received any Notice to Keeper through the post?