I don't think its possible to generalize about an entire area's attitude towards anything. I'm sure some people would be annoyed by your animals roaming, and others won't care.
Brunswick County has a law prohibiting animals from roaming free. You are required to keep them on your property. The county is also home to bears, coyotes and bobcats, so your cats probably wouldn't be coming home if they do roam.
Supply is a long way from the beach (via a golf cart). Here is the county ordinance on golf carts:
http://www.brunswicksheriff.com/site...-Ordinance.pdf
Apparently, it is legal to do what you describe, although I suspect it would be very difficult to find a route entirely on roads with a maximum speed limit of 35 mph. I think it would also be very unsafe.