I think what you're missing is that while the first pitch to B3 legalizes the previous at bats - it does not legalize B3! And if B1 was the previous batter ... B3 is not the correct current batter.

If B1 was the previous batter and B2 is on base and B5 stands up and takes a pitch, then B3 is the correct batter. If during B5's illegal at bat B2 scores and then the defense appeals that B5 is the incorrect batter, they are correct. The appeal is upheld and B3 assumes B5's count.

Hmmm... given just this scenario, by what rule are you putting B3 at the plate instead of B2? At the time of THIS appeal, B1 was the previous batter, and B2 is not on base.

I'll phrase the question this way: Can a legal batter be "illegalized" during her at bat? There is no doubt (at least to me) that B3 was legal when she came to bat. But does she become illegal the moment B2 scores (or is picked off, or steps off early, or whatever)?

Or, once she entered the batter's box legally is she entitled to stay?

I might conceivably call for B2 to bat if I did not know she was the player who just came off the bases, and I didn't go to the official scorekeeper to help me with the situation. But since I do check the official book when I work out a batting-out-of-order appeal, I would hope that the scorer would enlighten me to the fact that B2 was the runner who started out at third base. Then I would call for B3 to bat.

Remember, at the moment of the first pitch to B5, that legalized B1, so B2 should be the proper batter. But at that moment, B2 was on base, so she gets passed over and B3 becomes the proper batter. Subsequent play while B5 is up to bat doesn't change that fact, so an appeal after that first pitch to her has no bearing on the situation. There is no rule or interpretation that says a batter who has becomes the proper batter (in THIS scenario, B3) can suddenly become improper during subsequent play.

I'm not sure what "this scenario" is. If you mean mine, I'm putting B3 at the plate because she is the proper batter (7-2-c-4 in a fairly old version of the book). They appeal that B5 is incorrect. I confirm that B5 is the incorrect batter and since it was discovered while B5 was at bat I send B5 back to the dugout and bring out B3.
Think of it this way, the question is not: who should be batting? Rather it is who should have been batting. Let's say B5 instead of being caught while at bat is caught after hitting a triple which scores b2. Are you saying that since B2 has scored at the time of the appeal that B2 is out for letting B5 bat in her spot instead of B3. And that B3 is now the correct batter?

(Interestingly, the NFHS rule reads:
When several players bat out of order before discovery so that a player's
time at bat occurs while she is a runner. Such player remains on base, but
she is NOT out as a batter.
So in my scenario, since only one batter batted out of order such that b2's turn at bat came while she was on base, I'll have to call her out :D;):D
)

2-D-4 does not state any of this. 2-D-4 simply says that we do not remove players from the bases if BOO is discovered during an at bat. But if B2 is not on base when BOO is discovered, why would B2 not be the correct batter?

I think you mean the scenario where b2 bats for b1, b1 bats for b2, and then b3 comes up to bat. If so, NO B3 was not legal when she came to bat. The correct batter when b3 came to bat was b4. B1 had just batted out of order for b3 and since no pitch had been thrown, b4 was still the correct batter. B3 became the proper batter when she took a pitch because her name was the first name after the last proper batter (b1) who was not on base.

Because the legalization of B1 took place when the first pitch was delivered to B5, by 2-D-3a. And by 2-D-3c, the next batter becomes B2 at that moment, and at that moment, she's on the base. And by 2-D-4, she cannot be removed from the base at that moment, so B3 becomes the correct batter.

You seem to be hung up on when the BOO is discovered. That is immaterial. What counts here is when things become legalized, and that happens the moment the first pitch was delivered to B5. There is nothing that allows for a batter--B3 in this case--to start out proper and then become improper during an at-bat.

Guys this is really simple. Once a pitch is thrown to B3, B1's time at bat is legal (ASA 7-2-3). Because B2 is on base at the time of the first pitch to B3, B2 is skipped in the Batting Order (7-2-4) and B3 is now the legal batter.

And, because she is the legal batter there can be no BOO. The bottom line... We have nothing. "Sorry DC you should have appealed before the first pitch to B3."

I'm not "hung up" on it. When BOO is discovered is how rule 7-2-D is written. In fact, the first words of 7-2-D are "If batting out of order is discovered:" and then giving 4 subsections telling us what to do based on when BOO is discovered.

I find no verbiage to indicate that the placement of runners at the beginning of an improper at bat matters at all... just directions on how to determine who the proper batter is, and then directions on what to do if the proper batter IS ON BASE WHEN BOO IS DISCOVERED (the opposite of WAS on base PRIOR to BOO being discovered).

Honestly, what you're saying makes sense, conceptually. And it may be what the rulesmakers intended... it's just not what the rule SAYS.

Because the rules say a pitch to the next batter, not the next correct batter.

But 7-2-4-D doesn't say that. It doesn't say anything about where the proper batter might be when the first pitch to the wrong batter happened. It STARTS with "If batting out of order is discovered:" The whole rule - all four sections - hinges on when BOO is discovered.

There is no BOO so it does not matter when the appeal is made. B3 is the legal batter per 7-2-D-3 and 7-2-D-4. Play ball.

I mean ANY scenerio where the proper batter is on base and gets off the bases before the at bat is complete. That is the question asked by the OP in the abstract, without complicating it with names and numbers.

In your scenario I agree with your rulings.