We, or I guess I should say at least the umpires in my area are taught to leave the bat alone. The catcher knows where it is at and if the umpire moves it we could potentially move it somewhere that the catcher isnt aware of and cause more problems. I have also been told there was a local case where the umpire moved the bat and a claim was later filed against the umpire for damaging to the bat.
If there is nothing going on at the plate and the on deck batter can get out there, move it and clear the area, sure, get the bat and move it. But, more times than I can count I have had a runner coming down the 3rd baseline, a throw coming in and suddenly I have another helmet coming into view from the side to clear a bat. At that point, leave the bat alone and stay out of the way.
If there is nothing going on at the plate and the on deck batter can get out there, move it and clear the area, sure, get the bat and move it. But, more times than I can count I have had a runner coming down the 3rd baseline, a throw coming in and suddenly I have another helmet coming into view from the side to clear a bat. At that point, leave the bat alone and stay out of the way.