The answer is a very simple, "Yes, you do." Keeping a weapon in the vehicle is not bearing in the sense that it is carried by or on a person. The minute someone steps out of the car carrying/bearing a weapon on YOUR parking lot, then you have a right to prosecute, fire, or discipline, as you please. The parking lot is yours; their car is theirs. No private property dispute at all.
Declaring rights appurtenant to a car equivalent or even superior to a title to land is going a lot farther than you might have considered.