Ive learned a lot about the Bully AI system when I made an old mod (which is now united with Red's mod and were working on it).
While I have been testing it, Ive learned when ped's will do a running attack, when they will grapple, ect...
Ive also learned when they will shoot.
Ive noticed that my bodyguard dosn't shoot when im near an enemy.
I got a few steps back from the enemy and my bodyguard started shooting.
The reason for that is that R programmed it like that to avoid the bodyguard from hitting the player I believe.