Hopefully, your complaint to the airport will bear fruit, but don't hold your breath. It is good news that you haven't identified the driver although you may want to edit your OP as they way you have explained it, it is fairly obvious who the driver is, although no one but you knows their identity... for now. You are Keeper and you refer to everything that the driver did. NO "I did this and that". Only "the driver did this and that".
As you are dealing with a vexatious bunch of ex-clamping thugs, you are not going to get anywhere with this by appealing to either VCS or the IAS. The only way this will ever be resolved is if it goes all the way to the ultimate dispute resolution service, the small claims track of the county court, where a judge would decide whether the keeper owes VCS a debt or not.
Whilst you quoted the airport bylaws, you have not been issued a penalty notice under those bylaws. You have been issued an invoice for an alleged breach of contract by the driver. Very different thing.
The primary defence will be that VCS cannot hold the Keeper liable for the actions of the unknown (to them) driver. As the airport is land that is under statutory control (bylaws), it is not "relevant land" for the purposes of PoFA. PoFA is the stupid law that allows a PPC to transfer liability for the charge from the unknown driver to the Keeper, even if the Keeper was not driving or even in the car.
This little tidbit from VCS's response to you: "
...we will continue to pursue this matter on the reasonable assumption that you were the driver..." just goes to prove their intellectual malnourishment and vexatiousness. There is plenty of persuasive case law to put that silly argument to bed. VCS will not be allowed to infer or presume that the Keeper must have also been the driver. If you're interested, here is a recent appeal case with the same company that proves the point I'm making:
VCS v Edward.
There is no point trying an IAS appeal, even though others will disagree. With a less than 5% of being successful, it is up to you if you'd like to put ay effort into it.
What you will have to do is weather the stream of debt collector letters you are going to receive before they eventually (they may never) send you a Letter of Claim (LoC). If/when you get one of those, come back and we'll advise on a response and subsequent actions.