These direct trains are quite short, hence not many seats and indeed have mandatory RES summertime-plus that suddenly about each&every Swede follows a girl named Gretha and wants to go pa taget (=by train). You can try via site of DSB (danish trains), but I think they may not allow direct RES for seat only. They have even bought old stock from DB to lengthen the trains-but this must first be renovated etc-that east time.

Smarter people divide up on planner in segments-then you get to see ALL trains from A to B to C etc-

like Hamburg to Flensburg (lst stop in DE) -DSB till KOlding/Fredericia to Kobenhavn-DK also has a few overnite trains, seat only, for inland use. The planner of bahn.com will also usually show more options.

Kob-STO fe.-use local train till Malmö via tunnelbridge-from M many more trains onward.

I suggest that you travel to Copenhagen during the day of the 7th of August and then stay over night there. Then you have no problem catching the train to Stockholm in the morning of the 8th.

As far as I can see when I looked at bahn.com there is only one connection on the 7th of August with 5 seats available in 2nd class from Hamburg to Copenhagen, see picture below. You need to make the reservation ASAP to be sure to get the seats. Make the reservation at bahn.com and select "Seat only".

The IC398 can only be reserved via the DSB Udland Page of B Europe 

Here is a link to DSB utland. Here several trains are available on the 7th of August. The picture shows the evening and night trains.

https://travel.b-europe.com/dsb-rail/en/reservation-only#Outbound

Thank you, do you have some recommendations for the cheapest one night stays in Copenhagen? I was looking at Generator with 20% Eurail discount, I just hope they’ll let us do a check-in when we get there at let’s say midnight.

Copenhagen is an expensive place to stay… 
Late check-in at Generator should be no problem. They usually have bars open until long after midnight and 24/7 working hours… (of reception).

If you are on a really tight budget, taking the train (included in IR) to the airport and sleeping there with sleeping bag might be the cheapest option .

@Radek Tuček 

I hope that you saw the answer where there is a train leaving Hamburg at 23.56. I don't know if there is any possibility to sleep on the train or if it is only seats.

I have no knowledge of accommodation in Copenhagen. 

Yes, I was counting on the 23:56 train, but  wherever I looked, the train reservations were sold out.

Then they sold out during the day. They were available when I made the post. I got as far as writing email address for the reservation before I canceled.

Through which site? I tried to book the reservations since monday, but it says that it’s sold out, at least at ČD (české dráhy) and DSB, Didn’t try other sites, deutsche bahn said that it’s even unavailable to book this one.

For your return journey, search for connections at Flensburg to use the non-compulsory reservation trains, it usually requires 3 changes Copenhagen - Hamburg this way, you will have to leave earlier to make your connection in Hamburg.

https://travel.b-europe.com/dsb-rail/en/reservation-only#TravelWish

Checking this site too, but haven’t seen the train available for 5 people since monday. Guess I’m just unlucky.

Nevertheless, I’m trying to book the train at the 14th of August that leaves Copenhagen at 06:56, has a train change at Fredericia (we have just 7 minutes to get on a train to Hamburg) and arrives at Hamburg at 12:06, so we can still catch the train to Prag that leaves at 12:51. The problem is that Deutsche Bahn doesn’t let me book this train, it says that there has been an error and won’t let me pay for the reservations. It looks like I’ll have to book the trains through DSB, even though they are more expensive than Deutsche Bahn. 

I just hope that 7 minutes for train change is enough for the train at Fredericia.