The possibility to link this country to one from some other show is hampered by that trivial tidbit. O'Bviously it can't be any of the many fictional Balkan countries seen in TV shows like 'Mission: Impossible' or 'The Man From U.N.C.L.E.' as they would be found on Toobworld maps in the 1960s.
So I went back to a series set before 1915 - 'Sherlock Holmes', the version starring Ronald Howard that aired back in 1954. Unfortunately, the episode I had in mind didn't name any of their countries involved either....
The Case of the Royal Murder: As payment for detective services rendered, Sherlock Holmes and Dr. Watson are invited for a weekend in the Balkans. Their holiday is cut short when the visiting Prince Stefan is poisoned and their host is accused of the murder. Only Holmes can uncover the truth - but first he must avoid being the next victim.
(Description from the IMDb.com)
Prince Stefan was the son of King Johan from the neighboring country. It was Stefan's cousin King Conrad who was under suspicion for his murder.
I've always held that King Conrad's country was Caronia, which was featured in three episodes of 'Get Smart' and which had an ocean liner named after it. ('The Lucy-Desi Comedy Hour')
So even though I have no name for that second country, it will be my contention that it is the country that would one day be ruled - for a short time, anyway, - by King Dimitri.
More than likely, Dimitri's country was absorbed by neighboring Caronia after World War I and that's why it disappeared from maps after 1915.