I'm re-playing the game on Extra Hard, and I got to this point again a couple of days ago... I found it's actually easier to shoot down all the choppers if you don't use the emplaced gun. Instead, try the Ultima Weapon, the mightiest and meatiest firearm aside from the Death Penalty. In my case, it was Ultima Weapon + Faerie L Barrel + Power Booster Gamma + Cerberus Relief Gamma. Each DG helicopter went down in a maximum of four shots; and since you're not stuck in that emplaced gun, it's easier to run around the place and dodge missiles. I got all 15 helicopters, but I wasn't trying particularly hard so two WRO soldiers were hit by stray missiles.

Ultima Weapon is tons of ownage. If you don't already know how to get it, it's not too difficult - simply modify the Model Gun (found inside a nearby building after saving the boy in Edge) into its best form. Takes a lot of Gil, but it's well worth it.