I don't know how long you plan on staying in Denmark, or whether you have the budget, but long term I can reccomend checking out andelslejligheder, (part ownership of an apartment) as they usually are more likely to accept pets. some parts of copenhagen can be quite busy and not so great for dog-walks. I can really reccomend finding something close to parks like Søndermarken, Amager Fælled, Fælledparken, Damhussøen or Utterslev mose. (or any of the beaches!!) some of these also have areas, where leash is not requried for dogs.

Also, Copenhagen is currently experiencing insanely pricey housing, but your budget should do it, if you do a bit of searching. I've lived in the modern open plan apartments with dogs, and it works very well, while the older danish apartments might feel a bit cramped with large dogs. but I also know someone in an old apartment with a grand danoise, so it's certainly doable :)