South Carolina had a very strict conscription law, and did not automatically grant exemptions to physician, or so I am advised. You are of course correct that surgeons and assistant surgeons for volunteer units which entered Confederate service were actually appointed by the Medical Bureau, but they would customarily honor the State appointments. And the Bureau did sometimes weed out inefficient physicians.
The Richmond National Military Historic Park has its headquarters on the grounds of the site of Chimborazo Hospital, and the Park Service has created a small but very nice museum devoted to medical services both North and South. Given the fact that Chimborazo was the largest hospital in the world at the time, it is certainly an apprpriate place to put a medical museum. Anyone visiting in the Richmond area with an interest in Civil War medicine woould be well advised to pay the museum a visit.