I would suspect a mix of Italian, Turkish, and Greek. Even though large Ladino speaking Jewish populations as well as Armenians existed, I don't think they were as prominent in the trading life of the empire. In the old city (the peninsula), some combination of Greek and Turkish would probably have probably have been the common language, and a dash of some of the less prominent languages was likely necessary.

Across the Golden Horn in Galata there was an Italian trading colony. I would suspect if you were dealing with any trade leaving the empire you have to know some Italian to interact with them. Of course, this was the era of the actual Lingua Franca, which varied from region to region but was a mix of Latin languages, Turkish, Greek, and Arabic. Internal trade would have likely been conducted in a mixture of Greek and Turkish.

Lots of sailors were Greek speaking but administrators and custom agents would likely speak Turkish. I had a friend show me an archival document once from the early modern period, and the merchant was a Greek speaker who had to write a petition to the Turkish speaking administrator who allowed access to the Black Sea from the Bosphorus. The Greek speaker went by "One Eyed Giorgio," but the phrase he used for One Eyed was Turkish. The fact that Greek, Turkish, Arabic, and Armenian borrowed so heavily form each other should give you an idea of the commonly mixed usage of these languages in trade. Anyone would have to be at a minimum bilingual.

Also, somehow the first "n" in Constantinople got lost in translation- the Turco-Arabic name for the city was Kostantiniyye! قسطنطينيه, if you can read it.