I'm not an umpire, but I believe once the umpire calls a balk, the ball is dead and it doesn't matter if the pitcher actually throws the pitch and the batter put the ball in play.
in NFHS rules, a balk is a dead ball. You're correct on the MLB rule. The play is live and if the batter successfully reaches first, the team has the choice of the balk or the actual play that resulted. In NFHS, under all circumstances, a balk call immediately becomes a dead ball, so anything that subsequently follows is moot.
Team does not get a "choice of the balk or the actual play that resulted". The balk is either enforced or it is not. If all runners, including Batter/Runner advance one base then the balk is NOT enforced and players a liable to be put out trying to advance further. If there is an out prior to all runners advancing their one base then the balk would be enforced and any play/out would not count. Example: Runner at 2nd balk called batter his ball to 3rd and tags the runner coming from 2nd. BR is safe at 1st. R2 goes to 3rd on the enforcement and BR comes back to plate. Same situation (R2). Balk called batter hits ball to outfield. R2 thrown out at the plate, BR safe at 1st. Balk is not enforced and the out stands.
Meant to add for MLB rules the previous results are valid. For high school as it has been said before ball is immediately dead. So if a balk is called and a kid hits an HR, does not matter, ball was already dead.