Sen. Rand Paul wasn’t invited to speak at this weekend’s National Rifle Association annual convention because the Kentucky Republican is caught in the crossfire between competing gun-rights organizations.

Top NRA officials are unhappy that Mr. Paul has for years lent his name to fundraising solicitations for the National Association for Gun Rights, a group that fashions itself a more conservative alternative to NRA. Mr. Paul’s aides have been told by the NRA he will be unwelcome to participate at NRA events as long as he remains...