When airport cases pop up in the private parking forum here, the usual advice is that the byelaws mean that the land is subject to statutory control, and is therefore not 'relevant land' for the purposes of Schedule 4 of the Protection of Freedoms Act.
This in effect means the operator cannot recover any charges from the keeper.
That the existence of byelaws might preclude the formation of an enforceable contract against the driver* isn't an argument I've heard before, although IANAL.
*That might not be what the lawyer who was interviewed was suggesting, I'm always wary of drawing conclusions from newspaper quotes on these sort of things.