This time I've found out you can get your bike on the night train from Copenhagen to Basel. Likewise you can get it on the night train from Vienna - but only to Hamburg.
I have friends (I met in China) in both Alpirsbach in Baden-Württemberg and in Vienna.
That's why the trip will go to Offenburg on the Basel line. From there I will go by bike first to Alpirsbach and from there to Vienna.
It becomes obvious that it's a good idea to be VERY early with the tickets. The cheap tickets are long gone when I try a month in advance.. Another thing: you cannot make a reservation for the bike directly at DB (The german bundes train), but have to go through DSB, the danish train company. That increases the prize. Maybe you can get some german - if you know someone ready to pay - to make the reservation and clear the economic matter later?
The route from Offenburg to Vienna went along the Boden Lake, across the Arlberg pass to Innsbruck and across the Gerlos pass, along Tauernradweg to Salzburg and further on to Passau and finally along Donauradweg to Vienna.