Jeremy,
I have worked in Denmark for 10 years so know a little. There are no decent sites within easy distance of Copenhagen city centre. I was looking myself last summer while I was working in Copenhagen. All the good sites are in the West of Denmark and on Fyn. There are some good sites in the north and south of Sjaelland but not within easy commuting distance to Copenhagen. However, they would be OK for a day trip into the city. I assume you are on the Newcastle to Esbjerg ferry therefore there are some fantastic beaches just north of Esbjerg (Blavand, Henne)and some decent sites. Try www.dk-camp.dk to see just about all campsites. By the way Denmark is a great country and the people are really friendly.