I don't know if anybody is going to read this, or care or anything, but...
In For the Glory's AGCEEP mod, if the Ottoman Empire conquers Egypt, he gets a flavor event of him becoming Caliph by convincing the last Abbassid caliph to hand the title over. The problem is that such an event is today seen as a fabrication or a hoax, invented by later Turkish historians. There is no evidence that this has ever happened. I'm pretty sure the Ottomans have made sporadic claims of being a caliphate (and in fact, tried to use that to get some Muslim nations, such as Morocco, to recognize this), but did not really need the Abassaid caliph to hand that title over to it. Rather, it would likely be due to the Ottomans' conquest of Constantinople, Mecca, and Medina, that would be the reason why the Ottomans would make such sproadic claims. Even then...the title of Sultan was preferred. Thus, having that event where the Abassids formally transfer power to the Ottomans does sorta ruins the historical immersion, etc.
The real "date" for when you can say that the Ottoman Empire started continually claiming the title of Caliphate was in 1774, during the
Treaty of Kuchuk Kainarji. This treaty, signed after the Ottomans' defeat in Russo-Turkish War, gave the Russians power to intervene in Turkey on the basis of protecting the Othrodox population. But it also gave the Caliph the right to protect Muslims in Russia. Thus, this treaty was commonly seen as the first time when the Ottomans used that title of Caliph outside of its border.
When the Ottoman Empire was prosperous, it did not use the title of Caliph. But when the Ottoman Empire was declining, the Ottomans saw the title of Caliph as a way to regain its former prestige and power, by finding a way to influence European powers by claiming some status of leading the Muslim minorities. In this way, the title was useful to promote the Ottoman Empire, and so the Ottoman Sultans continued to use the title. The rest is history.
So, if one wants to be at least somewhat...semi-historical, you could tie the decision to become a Caliph based on the survival of the Ottoman Empire. If the Ottoman Empire is losing battles, then it could receive the decision to officially claim to be the Caliphate and lead Muslims, and try to regain its power through claiming a leadership role for Muslims.