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The improper batter's (let's call him B8) at-bat is indeed over; he became a runner. The other site is simply wrong if they're saying the offense is able to replace him with the proper batter without penalty, or that the balk is now enforced as the result of a BOT appeal. The result of the play has been determined by rule.
With regard to the balk, never mind the legality of the batter and focus on the outcome of the play. The batter and all runners advanced, so ignore the balk.
Now deal with the appeal: Upon proper appeal, call the proper batter (say it was B6) out, remove B8 from the bases, return R1 to first, and send the new proper batter (i.e., B7) to the plate with a 0-0 count.
This is easy if you think about what you would do WITHOUT a proper appeal: run scores, B8 is now R2, B9 hits.
The appeal only undoes the result of the play, which has already been determined. You don't have to do anything beyond ruling on the appeal.
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