Those NM regs are quite extensive! I've checked all states I'm planning on traveling through. Most read similarly to SC regs. The trick is how tightly states enforce it and which person is inspecting. I've read where several midwest states are very serious about CWD and will nab hunters on their way back east. Planning on making it as clean as possible to avoid any questions.
It is illegal to import into South Carolina or possess in South Carolina a whole cervid carcass or carcass part from an infected state (see list below), except the following may be imported:
-Quarters or other portions of meat with no part of the spinal column or head attached
-Meat that has been boned out
-Hides with no heads attached
-Clean (no meat or tissue attached) skulls or skull plates with antlers attached; antlers (detached from the skull plate)
-Clean upper canine teeth from elk, also called “buglers,” “whistlers,” or “ivories;” and finished taxidermy heads.