I would try checking your particular dates on matrix.itasoftware.com. It may turn up some connecting possibilities that you hadn't considered. Be sure to change the city of purchase to CPH so that it prices in Danish currency.

I agree that a rental car will be very useful in this area. Public transit in the Carolinas, where it exists, is clearly designed for people with a lot of spare time. Timetables are generally lists of times where you are guaranteed NOT to find a bus at a particular location. You should check one of the travel booking sites (www.orbitz.com, for example) for a comparison of rental car rates, which can actually be as low as $20/day plus tax.

Alternatively, this may be one of those situations where a good travel agent could save you some money.

Have a great trip!