so do you not farm or do anything in the game that requires you to pick up items? I wouldn't be able to play if I ran around at max inventory space on all my chars.
I don't play a lot of my characters, but if I do, I storage most of the unnecessary items until I'm finished. The characters I play regularly, are usually close to empty.
Kafra storage is a nice place to keep all your items. Try it.
All of my storages are too full, and since I did the quest like 25 times, that would be like a lot of stuff to move around for such a simple quest. I have like two characters on my main account that are not full, and none on any of my other accounts. I made room for like a weapon and the quest items; that was hard enough. I was just trying to get all my characters done, and having too many items to receive the hat didn't really cross my mind until it was too late.