Read your rule book carefully. It clearly states that "no run, (meaning zero, zilch, zip, nada), can score" when the third out of the inning is made by the batter-runner before reaching first base.

In this case, the batter-runner was out prior to reaching first base, because she opted to leave the field of play instead. So, how many runs can score? Zero.

Oh...and if you need a rule reference, check out NFHS 9.1.a. for this. I don't have my ASA, USSSA, or NCAA books with me at work. (Actually, I don't have my NFHS book here either, but I do have that handy dandy digital version available.)

You seem to have this one backward too...

You don't need a rule to not rule someone out. If you think this player should be out, there must be a rule that tells you this player is out for doing something. This player has not done anything to be called out for.

I understand what the book says. The problem here is that you have a league rule and it does change things. Whether it changes this or not is unspecified and I'm saying that as long as you have to make up the body of the rule, why not make it up in the most equitable fashion possible.
A solid argument might be made that we should be consistent with game ending procedures and I could respect that though I think it's less in line with how the game is actually played. The argument that the league rule can't change this rule but only other rules is somewhat lost on me though.

If the league didn't have a six run rule, you'd call him out right? 8-7-U?

To fork this thread a little since I'll be offline shortly:
To make this a little worse for your position, suppose that we have the bases loaded with no outs and the Batter gets ball 4. The runner from 3rd comes home and scores. Then the runners from 2nd and 1st (having legally advanced) run off the field (what with it being 6 runs and all). The the batter runner runs off the field. Run scores? If you have to make it up (and you do), make it up in the way it ought to be called.
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No one is saying that a league rule cannot supercede this. However, the OP makes absolutely no mention of that. (Again, no means no. None.) Barring that piece of evidence, we have to go with the book rule. And this book rule is as I have already stated.

Now, if the OP said that they "have a six run per inning rule, and that the runners who are forced to advance do not have to do so if the sixth run will score as a result of an award, and that no fourth out appeal, nor any other action by a player, coach or umpire can nullify a run once it has scored," I might be willing to entertain your argument. However, the OP never stated anything remotely close to that, and I am sure that no PARD director in their right mind (I know some who aren't) would ever make such a league rule.

You are really reaching here to defend an argument that has little to no merit and subsequently muddying the waters for those officials, players and coaches who might be reading this forum in hope of getting some concise answers.

No... I believe the problem here (and I hate local rules as much as the next guy) is that you're trying to say that since there's a local rule that might apply, then there must be other ones, written or unwritten, that we can pretend apply. There's no need to add additional rules.

The league rule CAN change other rules. They can, should they choose, specify that a batter in THIS ONE case doesn't have to go to first. But they didn't ... so why are you choosing to add a rule that doesn't exist?

Just because an inning (or game, really!) APPEARS to be over does not mean it's over. I have a coach that consistently tells his players to complete the play (on either side of the ball) in situations where the run-limit is reached or in cases where it appears the game is over. Several thought he was either crazy, or just trying to get in extra practice...

Until the one day, where he needed 5 (the run rule in that league) to win the game. 4 runs in, tie game, R1 on 2nd, R1 on first. 1 out. Ball hit to right. R1 misses 3rd and scores. R1 also scores and BR stops at 2nd. The other coach appealed the miss at 3rd and got the 2nd out - but since his team kept playing, they won anyway. Had R1 not continued, we'd have had to keep playing that game. I've seen this team get "outs" on defense after the apparent inning-ending run had scored, and so far it hasn't mattered, but it COULD, in much the same manner as above.

Of course the run scores. BR is out for failing to run to first. 1 out. 6th run scores, inning over. By what rule do you call anyone else out? And even if you do, by what rule would either of them wipe the run off the board?

This really isn't that complicated.

The only league rule we have for this age group is the maximum runs allowed per inning (6). Everything else is ASA. The explanations by mbcrowder and shatboi make perfect sense. Thanks.

I know a lot of times the things I say may make me sound like I have, but did you really need to point it out to everyone??? :( ;)

For the record, folks, my handle on here is a phoenetic pronunciation of "Scotty Boy," a name that was bestowed upon me by my late grandfather as well as several others since. Skah + t = Scotty, and boi = Boy.

8-7-U. I don't see why you want to ignore that rule. You're missing the whole thrust of my counterexamples because you refuse to confront that rule. And I don't really blame you because the league meant to have their six run rule supersede it but you can't get there until you're willing to admit you have to make up rules.
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VAPIR ONE 5.0

8-7-U does not apply to the BR

No but it does apply to the runners who start out at 1st and 2nd. If they run off the field for the first and second out before the BR gets off the field then if you call the BR out, you're calling her out for the third out (made before she reached first).
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Sigh... I already said the BR is out. She can't very well be out again, much less 3 times. I'm not ignoring 8-7-U at all.

So you are changing the play as offered in the OP? Nowhere was it stated that any runner entered the dugout without advancing other than the BR. That means the 8.7.U does not apply to the play offered.