My own two cents worth . . . in the scenario listed (dropped 3rd strike - batter has the opportunity to run to 1st), I've got "strike three" and "batter out" when the ball hit the bat (or vice-versa). Analytically speaking, the batter did, in fact, intentionally cause the contact of bat with ball. He/she consciously and intentionally dropped the bat for the express purpose of running to 1st. BR has an obligation to drop the bat in a place/location that would not cause interference with a possible play. Indiscriminate dropping or throwing of a bat that could alter the outcome of a play needs to be dealt with appropriately; in this case, by calling an "out" for interfering with the catcher's attempt to throw to 1st base.
Jerry
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