I have offered a solution to the conflict between private property rights and the right to self-defense.
Most large companies are corporations; that is, they are entities which owe their existence to the legislation which allows their creation and which endows them with limited liability. There is no "right" to form a corporation and doing so involves seeking permission from a government to do so and obeying the corporation laws within the jurisdiction permitting the incorporation.
I would see nothing wrong with adapting the corporation laws such that corporations do not have the private property right to infringe the right to self-defense in their parking lots.
Those who wish to conduct business without the benefits of limited liability can continue to enforce their private property rights as they see fit. If a person comes to harm because of the policies of such a private business, then the owners of that business will face full liability for the harm. Those who wish the benefit of limited liability will have to recognize that there is a price to be paid in exchange for that human-created benefit.