This worries me about Copenhagen. We have a 7:00 am flight on 7/1. I was going to rely on a taxi - should I look into something else?
I do not think you will have any issues with a taxi that early. People are not going to the port until many hours after you leave. It was kind of a crazy day for cruises. I think they had five cruise ships in port and I think they normally have four at max. What hotel are you staying at? The Tivoli is only about a block from the train station and the first stop is the airport. Fast and cheap. We may try this this year as our kids are older and better able to help with their luggage.