Instead, what if the state paid prisons a one-time fee for accepting convicts (the amount would vary depending on the type of crime committed), and then penaliz

Instead, what if the state paid prisons a one-time fee for accepting convicts (the amount would vary depending on the type of crime committed), and then penalized the prisons heavily for any recidivism after release? So you are proposing paying private prisons up front and then punishing them potentially ten to twenty years later. That doesn't sound workable politically nor does it sound like a good incentive structure.