There are a number of reasons why Michelangelo, or any Christian man would be hesitant of travelling to Muslim-controlled Constantiople at the behest of a heretic Sultan. In case it isn't totally obvious, Costantinople's conquest by the Ottomans had occured only a half-century or so previously. There were still people who experienced and even fought against the Ottomans in living memory, and the loss of what was once considered a bastion of Orthodox Christiandom was a massive sore spot even among the largely Catholic Italy. Michelangelo taking the Sultan upon his offer to design a bridge would invariably draw the ire of the most powerful man in Christendom, whichever Pope would be sitting at the time. Although admittedly there are times when Michelangelo may have considered actually taking the Ottoman Sultan's request up-- (the Sultan had left letters of credit in Florence covering Michelangelo's travel expenses should he ever decide to take up the Ottomans on their offer)-- most notably during the a period of financial difficulty between Pope Julius II and Michelangelo during the construction of his (the pope's) tomb. Michelangelo's own accusations that the jealous Bramante was manipulating the pope to attack the sculptor resulted in heavy friction between Michelangelo and His Holiness as his favors turned elsewhere, which eventually resulted in Michelangelo revolting and leaving Rome for Florence. It was during this time that Michelangelo briefly considered taking the Sultan up on his offer and expatraiting himself to the Ottoman Empire, although this would have turned the pope even more furious than he was already.

Eventually, the artist gave in and was forced to make a trip to the newly re-conquered Bologna where the victorious Julius II forgave Michelangelo in return for demanding that the artist instead cast a massive bronze statue of himself, which would take the artist another two years of his life. Unfortunately, this incredible bronze was destroyed just a few years later.

However, back to the question-- I personally believe Michelangelo would have been more flattered than offended, if at all. That a foreign dignitary would request the artist's prescene at all would have been hugely flattering and a point of pride for the artist, even if actually taking up the Sultan on his offer was largely out of the question. Among most things, that his predecessor/rival Da Vinci was considered before him (the claim of which I do not know) would have been secondary to the fact that Michelangelo would have been essentially going to work for the enemy.