WEAPON is actually pretty easy, as long as your people are level 88 or higher. Here is what I did. First, I Equipped Cloud with his Ultima Weapon sword, Barret with his Missing Score, and Nanaki with his Limited Moon. For their armor, I equipped Cloud with a Mystile, Nanaki with a Mystile, and Barret with a 4th Bracelet. For the accessories, I put a Speed Shoes on Cloud, a Championship Belt on Barret, and a Championship Belt on Nanaki. They each had their level 4 limit breaks. Here is what I equipped materia-wise on them (on their armor slots only, so they could gain that load of AP):
CLOUD: Master Magic/Mastered Quadra Magic; Mastered HP Plus; Mastered Phoenix/Mastered Final Attack; Mega All; W- Magic
BARRET: Gravity/Quadra Magic; Mastered Hp Plus; Mastered Mime; W- Item
NANAKI: Enemy Skill; Mastered HP Plus; Final Attack/Level 4 KOR; Mime
Here are the steps to defeat that pesky weapon. First, with Cloud, cast Demi (any level) at E.W., twice with w- magic. The resulting damage will be 80,000. Now mime this with your other two characters. Use a megalixer with Cloud if you need to. Use Demi with Barret, a megalixer with Nanaki, and then repeat this until he is dead.
Note: The Aire Tam Storm attack does damage equal to the amount of materia equiped, times 1,111. For instance, if you have 5 materia equipped, it will do 5,555 damage to you. Aire Tam is Materia spelled backward.