Stubborn Gifter

What the title said. I can't find my fluff after putting it to sleep. It said that it was "unequipped" at home. But when I went home, I didn't see my fluff. Try to "equip" at my Avatar dressing room, but I can't find the fluff to equip. Even with inventory, I can't "equip" the fluff. I can't even "equip" in Town either.

So does anyone know what to do when you put the fluff to sleep?