Can a rehabilitation physician bill to a private health insurer when they see patients at home as part of a hospital in the home program run by a public hospital?

I am a rehabilitation physician that works at a public hospital that offers rehab in the home and hospital in the home for patients, as part of early discharge or hospital substitution strategies. If a patient chooses to use their private health insurance while in a public hospital for the purposes of a joint replacement and is then discharged home for rehab in the home as part of the hospital the hospital substitution scheme - Can I charge the insurer a gap payment for my home visit to them and what item number should I charge a 116 (in hospital follow up) or a 128 (follow as part of a home visit)?