A baton is the stick runners pass to each other in a relay race. That's why the move is called this; it passes on things to the next player. It sounds like button. I'm surprised anyone doesn't know this.
Well, it doesnt sound like button at all... It sounds like - Bat - On. Baton.
But yes - I have heard it say in different games that its pronounced as Buhton pass. Without the A Like it should be