So as many have noted, it’s almost impossible to get a reservation crossing the border from Hamburg to Copenhagen.
I’ve done some research, and what’s really strange in my opinion is : the fact that you can board the train from Hamburg to Flensburg (closest station to the border) without reservation, and then you can get a reservation from Padborg (the first station on the other side of the border) to Copenhagen using the travel.b-europe.com (DSB) website. It’s the same train (IC 394), same schedule… But if you try to get a reservation for that same train but on any leg that crosses the border (e.g. Flensburg-Copenhagen), then you’ll get an error (from both Deutsche Bahn or DSB).
It doesn’t really make sense to me, basically the train is supposedly full just for the Flensbug-Padborg leg… It’s really strange.