about 6 years ago I was coaching a legion team using OBR rules. I had a runner at third working the line with one out. After the pitch, catcher tried to pickoff my runner, but he was back safely. This happened again. Eventually my lefthanded BR walked, and as he tossed his bat back toward our 3B dugout the catcher tried for the third time to pickoff my runner at third. You guessed it, the ball hit the bat and went toward our dugout. PU killed it and called my runner at third out. Was there an expectation that my runner would of been thrown out?? I don't know, but he hadn't the previous two attempts. There was no proof that this attempt was any different.
Needless to say, I came down the 3B line arguing that my BR had no intention of interfering with the throw. It was a freak occurrence. My BR was merely tossing his bat back toward our dugout just like most batters do when they walk. I lost the argument. So, I then said that he had no right to call my runner out, that if it was batter interference then my BR should be out. I lost this argument too. BR was awarded 1B and R3 was called out.

Now, if you was PU, how would you have ruled? Would your ruling change if the ball had rolled into DBT?

Time! BR to 1st, R3 stays. (Weak interference)

IMO

In the absence of an intentional act by the batter-runner, the tossed bat became inadvertently placed equipment.
There is no interference when the ball touches such equipment on LBT even if a fielder is unable to make a play due to contact with such equipment.
The umpire may then call time to prevent the inadvertently deflected ball from becoming a advantage to the base runners.
BR to first
R3 base TOP.

mick