I don't know if Tolkien's notes support this, but I always assumed that Treebeard's Entish name was something completely unpronounceable for anyone who isn't an ent, and "Treebeard" was a nickname that he picked for himself. Maybe because he finds it funny that other species think he looks like a tree. (I'm sure that ents look clearly different from trees to other ents.)

Edit: he says so himself.

Hrum, now, well, I am an Ent, or that's what they call me. Yes, Ent is the word. The Ent, I am, you might say, in your manner of speaking. Fangorn is my name according to some, Treebeard others make it. Treebeard will do.