The Ottomans were never particularly interested in making converts, preferring to rule with/through Orthodox/Armenian/Jewish elites, known as the Millet system. At times this also came close to recognizing various nations of the empire.
As far as I know, forced mass conversions only happened twice: In the Rhodope mountains of southern Bulgaria where the Sultan often spent his summers and desired to be served by Muslims, and in parts of Albania in order to stop/prevent the many rebellions there.
Thus, while the idea of inviting a large group of people to settle in the empire is not ASB at all, there is no reason why they should be made to convert. A very similar invitation took place when Spanish Jews were invited to settle around 1500 (most of them settling in Saloniki/Solun which had a Jewish majority up until WWII). A similar invitation would not be unlikely before the Ottoman conquest either. The area was regularly depleted by wars, and a few hundred thousand or more wishing to settle would probably be welcomed by local rulers.
However, all of this is likely to only produce a higher Roma population living in more or less the same conditions unless we can create an area of Roma farmers and villagers. Not to sound "Blut und Boden", but the presence of farmers create stronger ties to the land than other occupations, a sense of "this is the land of my ancestors" while Gypsies and/or Jews have traditionally lived more like eternal "guests" who will pack up and leave when conditions deter.