Great Question. I see two issues: 1) You'd have to have Islam develop some sort of territorial association with Religious Legitimacy, that is, descent from the prophet, religious knowledge, or piety can't in and of itself be considered to outweigh or necessarily be equal to claims of having the allegiance of the faithful of a particular region of polity, most likely Mecca, perhaps Jerusalem. 2) You'd have to have Muslim rulers, not actually there, and independent, accept its legitimacy.
In the end, I think you're best bet is some very early change, wherein one of the Prophet's male children survive, and is thus able to establish a literally and figurative Patriarchy. At that point, the concept of dualism might be introduced, where either the female (or likely male) descent from the Prophet could be considered "Political" and the other "Religious" Anywyas, a possibility. Without such dualism, however, such a figure as the Pope is unlikely.