Ok so it sounds like a silly idea at first, but check it.

So imagine this: Giant Earth Sprite spawns and it's go time, the mining companies across Eorzea are just tired of this bisnatch and they realize they are the only one that can do it, they have the true power! With pickaxe and sledgehammer in hand they venture forth in to battle!

So it would work in phases, phase one would be utilizing the use of a pickaxe to strike away the Sprite's outer shell while avoiding potentially life threatening AoE attacks and other bajungle. Once the shell is down it falls apart and screeches in pain!((That part optional, the screeching part.)) But on to the serious phase two, the sledgehammers come in and start beating the crap out of the core until it's nothing!

And the reward could be a bit of XP and some misc ore and what not.