I use Immortal Flames Sergeant Axe if I want max HP (with Sanction it's superior, because of it's immense AtkPwr bonus)
I use Garuda's Scream for when I want to trigger up my accuracy or when specifically fight her for the resistance
I use Ifrit's Axe for when I want max dmg output or when specifically fight him for resistance
I use Moogle Axe almost never, maybe only if I really could feel the effect of the "Change to regen HP"

The actual DPS's on the Axes aren't that much far apart. If you really want to increase damage output I would choose Ifrit's or Sergeant with Sanction.

I almost always use Sergeant Axe, guess Garuda weapons atm is just to show of that u won her more than once maybe :P. The Sergeant Axe plus some melded cobalt/sentinel gear and +1 jewelry get's my WAR on 4746 HP atm, I won't say I'm displeased with that ^^.