You don't trade them to the moogle, you trade them to the box, just like every other seal/weapon item. >.>

It only works for the trial you're on though, i.e. if you're on Briareus you can't turn in Sobek skins. But that's why I'm not doing Sobek yet; rather, I'm turning in my helms at 1-4 at a time since I solo them. xD Saves inventory space though, which is sorely strained!