I never knew that the hairstyle was a piece of head gear...? I would have expected that you got a consumable item the same way the hairstyles at the GS are distributed. If that's the case, the item itself wouldn't be the hairstyle, so you should still be able to move it to your inventory but you wouldn't be able to use it. I would think, anyways.

Regardless, the hairstyles from the lightning event are not head gear and are accessible only through the aesthetician. You wouldn't be able to throw it away, but if you were around during the time of the event and participated long enough to get one or two pieces of the gear set, then you would need to throw those out before you can take them from the mail box. If you've never participated in the event then it's definitely a glitch and I hope it gets fixed soon.