It came from the French word 'mousseron' and may be related to 'mousse', meaning 'moss'. The word 'mushroom' is, then, just a corruption of sound and spelling, having nothing to do with 'mush' or 'room', as you say.
So it can be in the song badger badger badger badger badger badger badger badger badger badger badger badger mushroom mushroom? www.badgerbadgerbadger.com for anyone who doesn't know what i'm on about! i'm gonna have that blooming song in my head all night now!