What should you do if you are bitten by a southern copperhead?

It depends how severe the bite wound is. Since non venomous species usually only have small teeth, any bleeding should be minimal. Wipe the bite area with antiseptic, and cover with a clean dressing to keep out germs. If the bleeding doesn't stop within an hour - see a doctor.
Wash your hands and the bite, rinse the snake body in cold water (this usually helps the snake to let go) and take allergy pills. Apart from that treat it like a normal animal bite.
you should still go to the hospital just to be sure, venomous or not they carry alot of bacteria in their mouths