Carrying the bat to first base isn't an out in ASA softball, so this rule is whatever your league says it is. By a straight ASA official rules interpretation, the run scores and there is no penalty.
Some softball rule sets do prohibit carrying the bat to first base. Two examples are NSA and USSSA softball. Under their rules the ball remains live, so the batter is out and the run scores (unless there are two outs when this happens- then the run doesn't count).