It should work, because the first train counts as a day and the second day starts in Hamburg. Maybe you have to "change" manualy trains in the app at a station in denmark after midnight, so that the app works fine. Or you make 2 trips one to Hamburg and one from Hamburg. Try, when you need help get here back. Maybe someone else knows more.

Sorry but I disagree - The journey from Copenhagen to Hamburg on day 1 will constitute the inbound journey as it ends in Germany, and the Hamburg to Stuttgart journey will not be a covered journey even if you have travel days left.

You can travel on as many trains as you want on your inbound journey but they all need to be started on the one pass day (from 0000 to 23.59 CET).

Why should this not work and Nightjet from Rome + ICE to Berlin Inbound works? 

Here the tutorial how it works. You need 2 travel days + one Inbound. 

In your case it should work without adding manually a trip, because the train stops in Denmark after midnight.

My interpretation of the inbound/outbound rule is as follows:

For 2 days of your pass your home country is included as a permitted country. Any pass day with a journey into, out of or within your country will be classed as one of those days.

Although these are designated as an outbound day and an inbound day, there is no obligation to actually cross a border by train on these days. Equally there is no obligation to use either of these days within your home country.

In the UK for example you could use a pass day to travel to an airport using any number of UK trains in the day (and even more trains in your destination leaving before 23,59. 

Equally there is no obligation to use your pass at all in your home country - I live 6 Km from my local airport and it can be as cheap to fly as pay Eurostar and TGV reservation fees. Then all my pass days can be used travelling on the mainland.

So if I am crossing back into Germany after midnight, then wouldn't my inbound day start then?

For reference, that's the planned journey:

The crucial thing is at what time your train crosses the border to Germany. Can you check that? Then you will get a correct answer.

Unfortunately it doesn't give me the option to check that, however as it's not long till midnight when the train departs in Denmark, and it seems to go all the way through Flensburg, I am fairly certain the train will enter Germany long after midnight. 

@Sarah Eberhard If you look at the answer from @rvdborgt in the link above you have the exact instructions how to manually add the trip from Köge to Hamburg in 2 steps so that your inbound travelday will be on the arrival day of the night train so that you can continue your trip in Germany on 22/8

day1: till Padborg, still in DK-last stop, no day in DE used..

Day 2-starts like 3-4.00-Pbg-zuHause, also mit nur 1 tag in DE. Du kannst in HH so lange bleiben bis letzter zug nach S mit BMW und Porsche gehen soll; Achtung heuer wegen Bombenentfernung in Göttingen-also an der Strecke- Gross-Umleitungen. NUr wenn du auch noch mit S-bahn nach Hause muss and dass >24.00 ist, dann musste zukaufen.