as a hat enthusiast and a rat, i sugest adding a classic top hat, you can't go wrong with that, or just put a rat on his head! It doubles as both a hat and a companion! :)
what about ducks?
ducks are heavier than rats so they may be more uncomfortable to wear as hats, but sure why not! Ducks))
well ants are WAY lighter....
(I included an image but it isnt working D: )
You could wear a whole colony of ants as a hat!