One good way to do this is in a large group. If writing letters doesn't work, then get about 20 folks together, who are all open carrying, then show up and ask to be seated. See what the response is.
You can add an additional twist by having a few pro-RKBA folks go in right before the "big group", and then once the "big group" is asked to leave, the "smaller group" (not carrying) then speaks to the manager and explains to them that they are leaving since the restauarant doesn't recognize RKBA.
Alternatively, you can have most of the "big group" take their seats first (unarmed), then have a couple of guys come in openly carrying who join the group. Then when the restaurant asks them to leave, EVERYONE gets up and walks out.
There are so many variations that you can have fun with. Many business owners only care about $$$$, so that is the language you have to use when dealing with them. Most will take down their signs when they start losing $$$$.
NC allows businesses to post prohibitions on concealed guns in their establishments.
A few years ago a Pizza Hut near here which I had often patronized suddenly posted a "no guns" sign on it's door. I immediately contacted a NC gun owner organization about the sign, and they in turn contacted the Pizza Hut franchisee. When I went by the place a couple of weeks later the sign was down. Frankly I was surprised, because the restaurant was doing a booming business even while it was still up.