For centuries, Christianity was virtually synonymous with Catholicism, and thus Christianity was a structured, hierarchical and patriarchal religion based out of a city in a different culture group from the origin point of Christianity. How can we get all that to apply to Islam? Bonus points if Arabia is not Muslim ITTL.
That's not true for much of early and Medieval Christianity, in the Patristic Age the Church was multipolar and Rome's primacy only came as a result of often being the mediator between disputes among the Eastern Sees. Islam's rise did much to weed out factions in Christianity, since the Miaphysites and Sees of Antioch, Jerusalem, and Alexandria were captured. However, most of the Christian population was in the Greek Church in the East.
If you want a more hierarchical form of Islam, then you're going to have to have the Caliphate's succession agreed upon instead of the Shi'i-Sunni schism. Shiite Islam is more hierarchical than Sunni because of the Sunni's primary source being consensus in the Umma, but if Ali is recognized immediately as Muhammad's successor than this could very well butterfly such differences.
Islam is rather biased towards Arabs in many ways, so it would require something like an invasion for a Caliph to be non-Arab, especially if it's done by descent of Ali (unless said Caliph is matrilineally linked to Muhammad).
However the Caliphates were already based in other cities, such as Damascus and Baghdad, so that is not a major issue.