If a CCW holder was under the investigation of a crime they would probably have had their CCW pulled by the sheriff already. I understand where the OP is coming from, not wanting anything that can be used against you in a court of law. But reality is, if a PA wants to put you behind bars they will dig up every little bit of dirt on you they can (did he kick his dog, spit onthe sidewalk, swear at his mother, etc.)
As for attending AA and being able to apply for a CCW, that would depend on which state the applicant resides in and what their states laws read. Here in Missouri, if a person shows a pattern of behavior, that in the opinion of the sheriff poses a threat to themselves or others, they can be denied a CCW permit. So the question then is, did the person have a pattern of behavior prior to application that would lead the local sheriff to believe that granting a CCW permit would not be in their best interest? Going to AA indicates a prior problem with alcohol consumption, was this documented as public record? If so, then receiving a CCW may be an issue (at least in my state). How about yours? What does your law say?