Had a border collie that would fly with us just about everywhere. Never really had an issue, from outback strips to capital city airports.
The only time we had a bit of an ‘incident’ was at Gold Coast airport whilst waiting to pick up my wife. We were talking a walk airside when two security guards drove up wanting to know why I had a dog airside. Explained what we were doing and one of the guards advised not to let him loose as they would have to shoot him! (Never saw the regs on that either!)