Soldato
It does seem bizarre. I believe the fight was completely random and unrelated to the reason he was carrying a gun.
This guy had a gun on him, and if the guy he was fighting with was actually his intended target, then there is no way he would have used his shots on an unrelated by-standing woman, he would have used them on the guy he was fighting. Which leads me to believe he simply wanted to kill anyone that day, there was no premeditated target.
Think about it, either get arrested for taking out your target or get arrested for killing a non-target.
It might have been a fight to stop him getting to her in the first place. Like you say, if you're armed and in a fight, why not use the weapon. All speculation really though. Need to wait for more details to be released.