NJ does not have a private school individual championship. They do have a "Non-Public" Team Championship. There are some 'prep' schools which do not partake in the NJSIAA tournament ALA Blair and Lawrenceville Prep, St. Benedict's...etc. But, as far as I know, they do not have their own state championship for those teams. If they do, it has nothing to do with the NJSIAA.
It's also important to remember that if you really think about it, it's only a few of the non-public schools that some people want separated. A large majority of the non-public schools in NJ aren't much better than the local public schools in their area, generally speaking. Quite a few of the non-public teams can't even field a complete squad. Hard to lump them into the same conversation as Bergen Catholic or Don Bosco (who used to have sub-par wrestling program until recently)