Please keep it Constantinople, allow change of capital
It was not just the moslems who referred to the Byzantines as "Rum". The emperor at Constantinople pre-13th century would probably have had you flayed if you did not address him as: "Dominus noster N. imperator perpetuus/semper augustus" (pre-8th century) or later in Greek as "N. en Christôi tôi Theôi Autokratôr pistos Basileus Rhomaiôn"... Moreover, when Baudouin of Flandres was crowned emperor (in Constaninople) in 1204, he took the title of "imperator Romaniae."
Hence: Romanorum > Rhomaiôn > Rum (for the empire of the eastern romans)
and the title of Istanbul being "Constantinopolis Nea Roma", I suspect the word Rum could occasionally be used to describe the city.
Pre-conquest common name among the Turks: Konstantiniyye
Other names used: Dersaadet (shortened version of the dar-ül-saadet thingy), Asitane. Does anyone know when the word Istanbul sprang up? It definitely predates 20th century but as to the etymology (is tin poli indeed), I don't know if anyone knows.
BTW, I distinctly remember once seeing an extremely coin of Mehmet II's bearing the legend: Caesar Romanorum et Sultanus Turcarum or some such. Minted in Istanbul, I guess. Does anyone have a gif or link?
But, otherwise, in communications with European powers: "serenissimus et potentissimus princeps dominus Sultanus Xxxx Han, Ottomannorum, Asiae et Graecia Imperator"