On the subject of carrying a handgun... as usual, an officer must have reasonable articulable suspicion to detain you. Though, remember that carrying a handgun in public, openly or concealed, is generally a crime in SC. LEOs can detain you if they have RAS that you're carrying a handgun in a place or manner that is not exempted by law.
SC has no "stop and identify" statute, so you have no legal requirement to identify yourself with phsycial ID or vocally if an officer detains you with RAS. From my research, you may detained for as long as it takes the officers to be able to identify you, though.
In a state that has a "stop and identify" statute (which SC doesn't haev), it is generally a crime to not identify yourself if an officer detains you and asks for your identity. The officer must have had RAS to detain you, though, in the first place.
Even though, SC is not a "stop and identify state", there are a few SC laws that make it a crime if you do not produce physical identification while performing a licensed activity. These activities include carrying a concealed handgun, driving an automobile on public roads, etc.
I am not a lawyer and this is the internet...