The on deck batter should not clear it, nor should the umpire. On deck would be altering part of the field, and the ump would be giving an advantage to one side or the other.
My daughter is a catcher and will clear it if she has time, keeps her from stepping on it and causing herself injury.
Also have seen umpires make a quick move to clear the bat when they can and the catcher doesn't. In either case I am glad they do for the sake of safety on all players parts.
The catcher should never touch the bat. That isn't her job. It was the batter's responsibility to properly dispose of the bat.
Is this rec ball and very young girls? I don't see an age group or reference to the girls ability.