Historically the Caliphs were more powerful than the Popes and more akin to caesaropapism of the Byzantine emperors.
The Caliphate reached its nadir only with the fall in power of the Abbasids with local emirs taking power. But even then they still theoretically acknowledged the authority of the Caliph. At worst, the Caliphs were like the Popes during their worst periods.
In 1519, the Caliphate switched from the powerless Abbasids to the Ottomans, and it became powerful again - far more powerful than even the Renaissance popes since they ruled an actual empire that covered most of the Sunni world.
I think your problem is that the Pope is just the Bishop of Rome. His authority is entirely spiritual, any temporal power is secondary at best. The Caliphs combine spiritual and temporal power because Islam doesn't distinsguish between religious and secular authority, but are combined. The Caliphs were always more powerful in both conception and fact than the Popes ever were. Your AHC "challenge" is already our world.
The only way to get the Shiites to accept the authority of the Caliph is to prevent the Shiite split to begin with. Ali would need to become Caliph, avoid any civil war, and survive so that his faction never diverges from the Sunni.