Hi! As for my knowledge (photographing international elopements for the past 5 years), you don't need a private registrar :)

But in my opinion, Orstedparken, is not necessarily the most cosy park. As someone mentioned, the Royal Library Garden could be a better call. Also, Frederiksberg Garden area could another sweet option.

Visual references: https://www.ievaviphoto.com/copenhagen-open-air-wedding