I am not a lawyer, and I didn't sleep in a Holiday in last night. But here are your answers based on my understanding of South Carolina laws which are linked at the end of this post.
1. SC does in fact recognize Michigan's CWP. Butin my opinion, if it takes any amount of time or money to get one, it's probably time and money wasted, because...
2. You can apply for the SC CWP as soon as you have an SC drivers license or ID card. And that's where it gets sticky. When you get your SC drivers license you have relinquish your MI license which means you're no longer a MI resident. If you're no longer a resident, then your resident CWP will become invalid.
Now someone here may know a way around this, but I doubt it. Because of my proximity to Shaw AFB, I've spoken to several incomingmilitary dependents who have found this to be the situation (active duty have an easier time). You also should know that after taking the mandatory class and sending off your paperwork the CWP takes 90 days to get. SLED will see to that.
That's just my two cents. Below are some useful links. Good luck with your move.