BigSky,
Just as an fyi, the privates drew up a proposal for a success based multiplier two years ago that would have moved them up if they had a prolonged period of success (ex: Bishop would have been playing in 2A playoffs this year). The state was actually the one that voted that proposal down after the committee presented it to the state, so it's not like there's been no attempt to try and level the field.
Ultimately, like I posted awhile back, I think the endgame for the state is going to be a separate playoff classification, like you suggested, where those schools (the Bishops and Lincoln Charters of the world) would compete in their regular leagues but then are reassigned to a separate classification come playoff time.
The problem with just doing it now is that there aren't enough of them to make it a viable division. However, with the cap on charters being removed recently, we're moving in that direction quickly. I would bet you'll see it happen by the 2022 realignment at the latest.
The caveat is that this isn't going to have the end result that everyone wants. It'll thin the 1A field of the charters and the parochials, but schools like Prep, which are considered true public schools (they just happen to be in an open enrollment county), are not going to get shifted over.