I’m so sorry your son is dealing with this. I’m regularly appalled at how poorly private schools handle bullying issues. One day they are going to be found liable.
I agree with earlier poster who suggested bringing it up with schools after your son is accepted. And I would absolutely make sure the current school states if/when asked that your son was in good standing and that it was a voluntary withdrawal for personal reasons. People change schools mid-year sometimes, just because family moved for example. What you want to make sure is that there is no misconception that there was an academic or disciplinary issue involved.