A private court of arbitration exists as a way for 2 parties who agree to use said court as a way to arbitrate a dispute.A more academic but easy to understand contextualization of the 2016 arbitration ruling from a Filipino academic.
China has not agreed to using any court of arbitration, whether private ones nor ones with international reach. Their position has always been bilateral negotiations.
While we can't discount the possibility that China may decide to bring SCS issues to a future court of arbitration agreed upon by both it and the other party, that's something for the future and not something currently in the pipeline (as far as anyone can know)