That's at issue here. Does the parking lot owner have the power to search my car parked at the airport lot at the beginning of the trip?
If your car is flying, then yes.
As I said, "Powers can impede or restrict Rights under limited and specified circumstance, but cannot abrogate those Rights." Take a cab; ride a bus; have a friend drop you off. If the parking facility has been credible warning of a potential act, and they take no action to mitigate that potential, would you consider that reasonable? Most folks wouldn't. By the same token, if you are the only one stopped, then you have a case for intrusion against your right to unimpeded travels.