Geoffrey De Smet mentioned in this comment
on StackOverflow that there should be a specific API for scenarios like a Dr.'s office when patients schedule times that are best for them. See that thread where I suggested some ways to rearrange the problem to make it more optimizable.
In my case, I have many constraints (e.g. the patient's insurance is in network, the insurance REST API says they have not met their deductible, etc.), and want to see all of the times available for this patient given the nature of the appointment.