The first work I consulted had very little if any information about missionary work in the early Ottoman Empire. I think the following quote, which relates to earlier Muslim expansion than Suleiman but is still relevant.

Nevertheless, though often less violent than has often been represented, the Muslim conquest was a major disaster for the Christian world. The ancient Eastern churches lost their dominsnt position in government and the world of thought. They were constantly drained of their resources through the defection of so many of their young men. (Neill, 55)

Though they were protected, a Christian, even an educated one could only rise so high in the administrative ranks without converting to Islam. There was also devsirme, which was the practice of educating Christian boys in converting them to Islam and bringing them up to be fit for the Jannisar Corps or the Civil service. Muslims did not kill the Christians or force conversions because they still needed people to work, but they could take away the power of the best and brightest of the next generation and take away the ability for the advancement of Christian ideas past a certain level.

I did find some information of interest in a book that focused mostly on the relationship of the Ottoman Empire with Catholic (Western) Christians in the lands under their rule. There was already an Eastern/Western split in the church which was both theological and political. In addition, some Eastern Christians were considered heretical by both the Orthodox and the Latin (western Christians so there was a further lack of unity. The Ottomans treated the branches differently as well.

The Orthodox [Eastern] populations of the Empire were organized by Turkish law into millets, or nations, under their own religious leaders, but Mehmet and his successors always treated Latin [Western] Catholics as foreigners. (Frazee, 7)

The Catholic or Western Christian areas that came under Ottoman rule had to make other arrangments for their freedoms which would make any missionary work likely precarious. So the Eastern Orthodox Church was depleted for various reasons and the enclaves of Western Christianity were in a precarious state. That leaves as you mention Christians from the West (Europe). However, the Church and political powers in the west were also distracted. As Frazee puts it,

There can be little doubt that Siileyman's success was in large part due to the particularly favorable conditions during his reign, which fell at the critical period of the Protestant Reformation, when Latin Christendom was divided into two warring factions. It also coincided with a time of great social unrest in Eastern Europe. ... Finally, the Ottoman sovereign was fortunate that the two strongest Catholic powers of Europe, France and the Habsburgs, were locked in combat for years, dissipating whatever strength a united Europe might have been able to gather to thwart his plans. (25)

Neill concludes that there undoubtedly was a Christian effect on Muslims, particularly Muslim mystics, but I am not qualified to comment or elaborate on that sorry! There was not however Christian conversion that one might expect from missionaries sent for that purpose. As for why there were missionaries to other relatively closed societies but not here research leads me to believe it is because there already were "established" churches even if they were greatly diminished in influence and numbers. The goal of driving out Islam militarily was still on people's minds rather than driving it out through conversion.

That said I did find one account of missionaries that went to Istanbul. Reproduced here since it seems to be what interested you the most.

The one incident which disturbed Catholic tranquility in the Ottoman capital at this time occurred when two over-zealous missionaries of the newly founded Capuchin Order [A 16th century reformation of the Fransicians] arrived in Istanbul in 1551. One was a Spaniard, Juan Zuaze of Medina, and the other a Neapolitan, Giovanni of Troia. Having met at Portiunelain 1550 they had decided, after consulting with the head of their order, on a mission to the Turks. Contrary to the practice of the Catholic clergy in the capital, they saw their mission as an effort to convert the Muslims to Christianity. Once the Turks became aware of what they were doing, the Capuchins were arrested, beaten and thrown into prison. The Catholic community bailed them out with the understanding they would leave the city.

The Capuchins sailed off to visit the Holy Land and afterwards went to Egypt where they imprudently sought once more to make Muslim conversions, preaching in front of the pasa of Cairo. The expected happened: they were thrown into prison and left without food and water. A week later the French consul at Alexandria came to Cairo to bargain for their release, but it was too late. They were found dead in their cells; the first Capuchin missionaries had become the first Capuchin martyrs. (Frazzee, 29)

Thanks for the interesting question! I enjoyed the research.

Sources:

Neill, Steven A History of Christian Missions, Second Edition Penguin New York 1986.

Frazee, Charles The Church and The Ottoman Empire 1453-1923 Cambridge University Press: NY 1983.

Edited to remove a statement that was not well supported. And to add clarity.