I wold not recommend stuffing a dog bed with cotton scraps. The electricity you would burn to get it dry after washing it is really brutal!
I don't like removing the egg-crate mat inserts because I think they are icky and should get washed too!
I vote for the SEALED fake-sheep-wool crate mats, but be sure you get the really heavy duty stuff that can't get chewed to bits. The whole thing can be thrown in the wash. It worked for Akitas and is working fine for our lab.
The fact that the fake sheep wool attracts pet hair is GOOD. It means there is less pet hair rolling into balls like tumbleweeds around the kitchen floor when they are blowing their coat.