This won't work.
The case is not moot, even if the new Board will change the policy, because the next election could result in a change back. The only way the abandonment of the ID policy could be made permanent would be if the Board consented to a court order to that effect-- but that would require that the court take action, so the court could indeed award the plaintiffs their legal fees.
Didn't the stickers end up in the school books and in front of students? Or did the case short circuit actually exposing students to it?
If students saw it, then damage was done and someone must pay, even if the board changes the policy.
And what about the disposition of all those creationist textbooks in the library?