I tried to hypothesize different scenarios on the duration of the Ottoman domination in Sicily:
1) Worst Case: 1571/1572 As a consequence of the battle of Lepanto, logistically Sicily is a quite realistic target for the Holy League. Consequently it could become a personal domain of Don Giovanni of Austria, who by marrying some local noblewoman, could give rise to an autonomous kingdom of Sicily, or be incorporated into Spain
2) 1647, with the great Sicilian revolt: the problems of Sicily at the time, from the recessive effects of the plague and climate change, demographic growth, the restructuring of agricultural property, remain unchanged whether the island is managed from Constantinople, either from Madrid, with the aggravating circumstance in this scenario of religious motivation and the interest of Spain and France in supporting the insurgents, for control of the island. Now, either Sicily becomes a dominion of Madrid or Paris, or it becomes a autonomous kingdom
3) War of Crete: for Venice and for her Italian allies, Sicily becomes an attractive target, to put diplomatic pressure on Constantinople. Obviously, it is difficult, due to structural problems, for the Serenissima to keep the island for long
4) Austrian-Ottoman Wars: as above. Now, despite the complaints of the Sicilians of the time, who were traumatized by Vienna's ahem strange demand that its subjects pay taxes, it was overall positive... A longer control would reverse the island's economic recession
5) Early nineteenth century, French and English occupation, due to the Sulfur issue
6) best case According to the nineteenth century, the process relating to the unification of Italy.
The longer the Turkish domination lasts, the more the Sicilian demography changes: a Muslim minority is created, given what is happening OTL will be concentrated in the area of Mazara and Trapani, the percentage of the population of African origin will be quite substantial and the Jews will return to Sicily very first.
However, the more I think about it, the less feasible a management of Sicily similar to that of Bosnia or Iraq seems to me, both for cultural and religious reasons.
Cultural reasons: the Ottomans find themselves faced with a strongly Catholicized society, with its taboos, rituals, particular cults, in which religion marks the calendar and social and economic relationships. A society that at the same time is highly militarized, due to the infinite local causes. If they try to put their mouth on religious issues, forbidding processions or the ringing of bells or local festivals, which also have an economic value, because they are also the exchange markets between the different provinces, they risk a revolt a day and Sicily becomes more than a source of income an expense.
Social and political reasons: the Sicilian organization is highly articulated and polycentric. For example, already in the fifteenth century the power of the brotherhoods developed, which in addition to the religious nature, act as trade unions and trade associations and mutual aid societies. An Ottoman governor will have to take this into account, unless he wants to manage indefinite general strikes (which OTl have jumped several Spanish viceroys). Or confront the great religious orders, which also have considerable economic power and which, if they are unwilling to pay taxes to their fellow countrymen, let us think of those they define as infidels. Or to the nobles who literally have their own private armies, or to the numerous autonomous citizens, given the Sicilian urbanization... Or to the question of the Sicilian Parliament or of the Palermo city administration: abolish them or put their mouth on them, as the Spaniards or the Bourbons is the best way to unleash riots to the bitter end.
Therefore, to survive in this asylum, in Constantinople they have to invent a new approach, delegating as much as possible the control of the territory to the many local power centers (also accepting their tax privileges)