Having a concealed carry permit only helps if you live in a state that has NIX Exempt. Alaska did when the program started, back in the early 90s, but did away with the NIX Exempt status in about 2005 or 2006. Dept Of Public Safety claims they do not have the manpower to run the checks. That was the major issue Sara Palin fired Walt Monaghan over, his refusal to hire more people. Currently we are having to undergo all checks just like everyone else.
Back when we had NIX Exempt status some stores refused to accept it (Sportsman's, Sports Authority, and Wall-Mart) and did the background check anyway.