MODIFY your post, the driver drove, the keeper received the notice and is now the one looking for advice!
There is no keeper liability, so you as keeper have no liability only the (unknown to them) driver, so best not to help them out.
"I am the registered keeper of the above vehicle and am in receipt of the PCN you issued. The land in question is subject to byelaws and as such it is not relevant land for the purpose of schedule 4 of the Protection of Freedoms Act 2012. Therefore there can be no hirer/keeper liability.
There is no legal requirement to identify the driver at the time and I will not be doing so.
I now expect you to confirm that no further action will be taken on this matter and that my personal details will be removed from your records"