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As I am not familiar with the Danish asylum procedure, I have to rely on my research in this case:
As far as I found out you are obliged to live in a reception center and later on (while your case is being processed) in an accommodation center. Since you said that you are still in the asylum process I assume that you are in an accommodation center at the moment, right?
Can you tell us what type of documents you have right now? Maybe it says something written on it concerning your rights to travel or obligations to stay in a certain place (like an accommodation center). If you tell me I can try to do some further research about it.
In the meantime here are some other ideas that came to my mind:
Maybe you can try to get a travel permission by your competent aliens department if your current documents don't allow you to travel abroad.
Another thing to keep in mind is that your girlfriend is German and therefore it's easier for her to travel to Denmark than vice versa. If it's hard for you to come to Germany maybe you two should consider to get married in Denmark. Or are there concrete reasons why you want to marry in Germany?
If all the options above fail, you can still wait for your case to be decided, since you will receive a passport along with your refugee status. If you got a high chance of getting asylum that would be the most convenient way since you just have to wait, but it will take longer of course.
I hope this answer was helpful to you!