Yes, this is a hill worth dying on, and if the VP has dealt with them before, you probably aren't going to die.
Not trying to sound dramatic, but the integrity of your program is at stake. If you yield, then more parents will try and "influence" the make-up of your bands to the point that auditioning students for placement (I'm assuming that's what you do) becomes a moot point. If you yield, what is this mom going to pull next in order to get her way the next time? She's already lying through her teeth. It sounds like she's already brow-beaten the counselor into submission, so it is up to you and the VP to put a stop to this nonsense. I would, calmly, put it to her like this, "You told me last year that he had to be in the top band so he could also be in athletics, he did not participate in athletics, and had no intention of doing so. His playing skills do not warrant him being in any other group than the bottom band. It is going to be like this: he will be in the band that is best suited for his skill level...the bottom band. The other option is for him to be scheduled out of band entirely. The choice is yours."
This IS the nuclear option, but at the same time, do you really want to have to put up with her?