You can make reservation on this connection. You just might need to split your journey in two, when reserving. Berlin-Hamburg and Hamburg - Copenhagen. 

For Berlin-Hamburg seat reservations are only optional, so not really needed. You can just board any train you want and find a free seat. 

All steps for the reservation of the Hamburg-Copenhagen trains are to be found in the link below:

@Braian Cuiña @BrendanDB Only in Summer there is a compulsory reservation for trains Hamburg-Copenhagen. If you don't travel in Summer you can reserve this routw directly by bahn.de online. You pay for the hole journey a reservation cost of 4,50€ in second class. Reservation is not compulsory, but reccomended on the trains to Copenhagen.

I booked a number of trains together with one change in Netherlands, through Germany with 2 changes and on to Copenhagen on the DB site yesterday with the seat-only option on the DB website so it’s possible alright.  There was only one booking fee - I think it was €5.50 - for all the trains combined.

If it will book somewhere in NL to Hamburg to Copenhagen this way, then I’m sure the DB site will be able to handle Berlin-Hamburg-Copenhagen as well.

Have a look here for how to book this type of reservation on bahn.de.

Yes 5,90€ is it for first class.

If you book via Interrail.eu or other companies, you pay for every train the reservation.

Via Interrail is it very expensive:

6€+2€ booking fee = 8€/seat/train