Seems like a simple fix would be to put an accelerometer into the key fob and turn off the radio if it hasn't moved in a minute or two.
Light can move about 300 meters in one millionth of a second. It's easy enough to find a store where the parking lot is well within that distance from the middle of the store.
So whatever system that goes into place would need to possibly have accuracy down to the 1us level or smaller.
1 nanosecond is about a foot. A 100MHz clock gives us a resolution of 10 ns, which translates to 2.25 meters away by light in air.
As a backup if your fob battery is dead, you could use inductive power in the door, or just use that to begin with (still with distance-bounding).