There is an auto train operating during the summer months (late April - first week of November) between Austria and Turkey - it is called the Optima Express. Auto trains "usually" take motorcycles, but the first thing you need to do is confirm this, and also determine if there is a lower price for a moto than the price for a car.
I have no knowledge of this train, I found the link above by using Google search after reading your post. It operates from Villach Austria (not Klagenfurt) to Edirne Turkey (not Istanbul). Edirne is the first major city inside Turkey once you cross from Greece into Turkey - it's way up in the northwest corner of the European side of Turkey, very close to the Greek border.
I suspect that the bulk of the passengers are Turks who work in Austria or Germany and return home for holidays in the summer.
Usually these type of auto trains are very expensive for a person travelling alone. They start to make economic sense when you have 1 car and 4 or 6 people who occupy a 4 or 6 bed sleeper room.
Michael
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