Yes. There are multiple grounds on which to challenge this - the airport being subject to byelaws means that they cannot use Schedule 4 of the Protection of Freedoms Act to hold you liable as the hirer, and even if they could, as you rightly note they have not provided the requisite documents.
At this stage you could appeal as the hirer on the lines of the below:
As Gatwick Airport is subject to byelaws, it is subject to "statutory control", and as such, not relevant land. You could therefore appeal along the lines of the below:
Dear Sirs,
I have received your Parking Charge Notice (Ref: ______) for vehicle registration mark _______, in which you allege that the driver has incurred a parking charge. I am appealing as the hirer of the vehicle. There is no obligation for me to name the driver and I will not be doing so.
As the Gatwick Airport North Terminal Drop Off Zone is not "relevant land" as defined in Schedule 4 of the Protection of Freedoms Act you are unable to recover the charge from me, the hirer.
I am therefore unable to help you further with this matter, and look forward to your confirmation that the charge has been cancelled. If you choose to decline this appeal, you must issue a POPLA code.
Yours,
If appealing online, make sure there are no drop down boxes etc. that force/trick you into revealing who was driving - you are appealing as the hirer, and saying nothing about who was driving.