He is committing a trespass the second he steps past a sign. While the law says he needs to be told verbally before legal action can start, he has committed the trespass-and like it or not, can be permanently banned from the property as a matter of public record. Get the letter from the cops, and it moves up to being a crime.
You are arguing your personal philosophies against the law. You can and will lose if push comes to shove. You maintain your right to self defense by not passing the sign. No one forces you to cross the line.
https://www.law.cornell.edu/wex/trespass
The gun sign clearly says you can enter, but not with a gun. Owners rule.
The Legal Rights of a Business to Ban a Person From Their Property | Chron.com
Bill of Rights controls government actions , not the small business man or even the big box. Gun carriers are not a protected class when it comes to entering private property.
Unless your state deliberately orders an exception, carrying past a sign is wrong. The laws I read provide an escape for the carrier if he misses a sign and requires a verbal warning. But the trespass starts when you pass the sign, and you are in the wrong legally.
I live in a small town, and if I start getting banned, I have to start shopping out of town. You do what you want, but you are wrong if you ignore the signs. Argue it any way you want, you are wrong when you carry past a no gun sign. And rude.
If you try arguing that you are obeying a higher law, you will discover that written law will prevail.