I'm taking inspiration from ImperialVienna's mention of the Spanish Byzantine rights purchase. Let's suppose the Spanish Habsburgs wangle that into their Spanish reign deal. My version takes alot less dead men, and IMHO's likelier because the Balkans and Contantinople->Istanbul have always been tough nuts to crack. It's alot earlier.
Some Habsburgs, including Philip II and John of Austria, visit ex-Constantinople, now Istanbul and the Ottoman capital, to check out their seat of empty power. They're enchanted and impressed, of course - it's quite an impressive city, having ruled alot of turf for a millenium, and been important and rich from its position well before.
Later, John of Austria gets the idea of taking Istanbul, offering to Phil 2 to take it if he can become the city of Byzantium's lord.
John plans to take the world's biggest defenses by surprise, using the big and powerful Spanish Navy, and letting it be known their objective was Venice. It works; Constantinople falls - still killing 344 Spaniards from its defenses, but the Ottomans are too surprised to effectively oppose.
Phil does make John Duke and Mayor of the renamed-back Byzantium. The Ottomans return to their prior capital.
Since Byzantium was all there was to the Byzantine Empire for a long time, that's all I need to reach your stated objective, all before 1889 ;-).