Unfortunately neither right trumps the other, as they are both fundamental. So though it sucks, the simple solution is to avoid places that will not allow you to defend yourself. I understand that sometimes you simply have to go somewhere that doesn't allow firearms, but usually you can get your business done at these places quickly and be on your way. While you're in there talk to someone about it. Maybe you can change some minds.
Also realize, many businesses lease the space for their storefronts. If they are calling themselves gun free, ask who put the sign up. Because if it wasn't the property OWNER, that sign doesn't mean much. Besides if you are properly concealed, no one should ever notice anyway.
Yes I am guilty of carrying in places that I am not supposed to. But I feel that it is my RIGHT to protect myself, and as a citizen it is my responsiblity to protect everyone around me. Even if they don't agree with me, or even despise me for my choice.