Equipping Lightningstrike weapons is helpful, due to the Extractor being a machina. If Tidus and Wakka can cause over 500 HP of damage to the machine, it will cancel its Depth Charges attack and come back down. Using Provoke will also cancel the charges, but will raise the Extractor's Strength for the next two rounds.
Extractor's Magic stat, which is used in determining the damage of Aqua Shooter (but not Depth Charge), is altered if Provoke is used. It is immediately increased to 30 upon provocation, but reduced to 20 before Aqua Shooter is actually used. When Provoke wears off, it is reduced back to 15.