Under Little League rules, with a runner on 3rd and 1 out, the game is tied in the bottom of the 6th with the home team at bat. Two strikes on the left-handed batter.

The pitch comes in and the batter swings late, strike three for the second out. The catcher bobbles the ball and on the batter's follow through the bat hits the ball, sending it toward the first base dugout.

R3 heads for home, but I call interference on the batter and return the runner to third.

On the first pitch to the next batter a long single into the gap scores R3, ending the game.

Two questions. Although the act was clearly unintentional, was interference the correct call? The rules that appear to apply are 6.06(c), 7.09(d), and 7.09(f) (but 7.09(f) seems to be a stretch).

And what would the correct penalty be? I can't call the batter out since he's already out. And all the rules covering batter interference have the batter out if there's two outs and the runner if out if less than two are out. Sending the runner back to 3rd just feels wrong. If it weren't LL, the batter wouldn't be out on the third strike since the catcher hasn't caught it, so he could be called out for the interference.

OBR Ruling
 

I realize your question specified Little League rules, but I thought I would provide the OBR ruling for those who do not do Little League.

6.03(c)
If a batter strikes at a ball and misses and swings so hard he carries the bat all the way around and, in the umpire's judgment, unintentionally hits the catcher or the ball in back of him on the backswing before the catcher has securely held the ball, it shall be called a strike only (not interference). The ball will be dead, however, and no runner shall advance on the play.

Absent any rule to the contrary, I would extend this ruling to Little League. It appears you got to the correct ruling (batter out and return runner to 3rd), although the reasoning might not have been exactly right. Had there been less than 2 strikes on the batter, he would have remained at bat with a strike added to the count.

P.S. I am having trouble picturing the ball going toward the first base dugout with a left-handed batter. Could you describe that in more detail?

I'm really not sure exactly how that happened. As soon as the the bat came around and made contact I glanced toward third to see the runner and in the meantime the ball had squirted down toward the first base dugout. Most likely the ball came off the bat, hit the catcher's glove or gear and went off the other direction.