Lebowski said:
The best solution is to have an event, that change name from Constantinople to Instanbul after few years.
About New Amsterdam/New York example.. I dont agree.
New Amsterdam/New York City became important and well known around the world after the Brittons captured it and made it world's metropoly. So using New York City for its name is more suitable.
But Constantinople was much more important as Capital of Empire than under Turkish rule. So in my opinion Constantinople is better name than Istanbul.
Game starts with the fall of the idealized romantic empire of your fantasies.

So it doesn't make much sense to argue it is any more important. However, it's true that the European world kept using the name Constantinopolis for the Ottoman capital, and so did the Turks in their own language (Kostantiniyye).
Constantinopolis (Greek: Κωνσταντινούπολη, Konstantinoúpoli, or Πόλη, Póli) was the capital of the Roman Empire between 330 and 395, the Byzantine Empire between 395 and 1453, and
the Ottoman Empire between 1453 and 1923. Strategically located between the Golden Horn and the Sea of Marmara at the point where Europe meets Asia,
Constantinopolis was extremely important as the successor to ancient Rome and the largest and wealthiest city in Europe throughout the Middle Ages; it was known as the Queen of Cities (Vasileousa Polis).
The city had many names throughout history. Depending on the background of people, and their language and ethnicity, it often had several different names at any given time; among the most common were Byzantium, New Rome (Nova Roma), Constantinople and Stamboul were some (see etymology).
İstanbul was the common name for the city in normal speech in Turkish even since before the conquest of 1453,
but in official use by the Ottoman authorities, other names such as Kostantiniyye were preferred in certain contexts. Thus, Kostantiniyye was used on coinage up to the late 17th and then again in the 19th century. The Ottoman chancelery and courts used Kostantiniyye as part of intricate formulae in expressing the place of origin of formal documents, such as be-Makam-ı Darü's-Saltanat-ı Kostantiniyyetü'l-Mahrusâtü'l-Mahmiyye.
In 19th century Turkish bookprinting it was also used in the impressum of books, in analogy to the foreign use of Constantinople. At the same time, however, İstanbul too was part of official language, for instance as part of the titles of the highest Ottoman military commander (İstanbul ağası) and the highest civil magistrate (İstanbul efendisi) of the city. İstanbul and several other variant forms of the same name were also widely used in Ottoman literature and poetry.
After the creation of the Turkish Republic in 1923, the various alternative names besides İstanbul became obsolete in Turkish. In an edict of March 28, 1930, the Turkish authorities officially requested foreigners to cease referring to the city with their traditional non-Turkish names (such as Constantinople) and to adopt İstanbul as the sole name also in the foreign languages.
Now the question is, are we going to use the local official name for most kingdoms, or are we using the plain English names?
Constantinople spelling is only used in English and in no other language.
For a fun reference, please note the names used for Istanbul:
Kōnstantinoupolis (Κωνσταντινούπολις) - Greek
Kostantiniyye (قسطنطينيه) - OttomanTurkish
Miklagord - Swedish
Qushta (קושטא) - Hebrew
Tsarigrad - Russian
Caergystennin - Welsh
Constantinopla - Spanish
Konstantinápoly - Hungarian
Qüstantiniyya - Azeri
Bolis (Ստամբուլ) - Armenian
Cařihrad -Czech
Constantinopel - Dutch
Ţarigrad - Romanian
al-Qustantiniyah (القسطنطينية) - Arabic
Costantinopoli - Italian
Mikligarðr - Old Norse
Konstantinopel - German
Kayser-i Zemin - Persian
Carogród - Polish
Stamboul - French
Mikligarður - Icelandic