It seems illogical that a hospital would ban bicycles, but you never know what some insurance company comes up with in it's tables (hmmmm...yeah, cyclists have accidents and get sent to emergency rooms....but so do motorists!). And since it's private property, they can do what they choose.
Still...a sign so stating their policy would be nice. Then, you can hold up a picture of that for public scoffing that a HOSPITAL would ban bicycles.
Also, I'd take issue with the fact that PUBLIC transportation is servicing PRIVATE property....a private property that bans a legal and legitimate form of transportation. I don't imagine that the local mall or shopping plaza would have a similar ban and still have PUBLIC transportation drop folks off at their front door.