You certainly can try to appeal - since at this point the claim is the patient's responsibility, you'd be appealing on behalf of the patient. I would just stick to the facts and explain to the insurance why it was necessary for the patient to receive these services out of network, that the service was ordered by an in-network provider. I agree that you shouldn't try to say that this was emergent as that could be taken as a misrepresentation, but you could argue, if the documentation supports it, that this was a necessary service that was part of the care plan and couldn't be missed and that no other provider was available that day.
It may or may not work and if not you'll have to decide if you want to require the patient to pay this bill. This is really an issue your front-desk staff or schedulers should ideally catch in advance. If it's known that the patient has no benefits, they could reschedule the visit or try to get prior authorization for out-of-network benefits, or at the very least make the patient aware that the visit won't be covered and that they need to make arrangements to cover the costs if they still choose to see the provider.