I just threw this together so be kind. Feel free to comment, change, rearrange, or add.
490AD: Relations between Theodoric and Zeno (Emperor of the ERE) falter and become sour (after an amicable agreement that arranged the overthrow of Odoacer Emperor of the WRE). Not wanting to enter into open warfare Zeno opts to further drive the unhappy population of Italy into open revolt. Claiming to be protecting the Faith Zeno moves the Papal seat to Byzantium. In part the maneuver worked as Theodoric was faced with periods of revolt not only along the peninsula but also through his power base of Gaul (his political standing tended to waver as he tried to gain the favor of Latins and Franks by adjusting the criminal code to favor one population over the other).
530AD: Schism of Pope Mercurius I (OTL John II) (the dual papacy of Rome and Byzantium). Mercurius’ pontificate began by gaining a broad standing approval from Athalaric (king of the Ostrogoths) and nobles – who were largely still subject to the whims of Byzantium. Mercurius began as a priest but his rhetoric about the damage Byzantium had done to the faith quickly gained favor with high and low born citizens (at the time there was a large Monophysit following in Byzantium which John and many others felt was corrupting the true nature of the faith). He also was a propionate against simony a recent edict by the Byzantine Senate. Using Mercurius popularity Athalaric promotes Mercurius to Patriarch of Rome and issues and edict stating that Rome was the true seat of papal authority.
580AD: Sole papacy restored to Byzantium. Over the last 50 years there had been several small conflicts between the Ostrogoths, their allies, and the Eastern Roman Empire (under Justinian I many of the forts and defenses along the Danube had been strengthened). During the latter decades of the 6th century though Slavic tribes began to migrate into Balkans. The forts along the Danube were able to repulse many of these attacks and force the bulk of the migrating tribes into Gaul and Italy. These waves of immigrants help bring to an end the Dual Papacy as Papal authority is resolidified under the Byzantine Patriarchs.
800AD: Most of the warring western kingdoms are brought under one flag by Charlemagne. Aiding in his rise to Emperor was the backing of Paschal, a Roman clergyman of great popularity. Upon his coronation Charlemagne proclaims Rome to be the only true seat of Christian authority and that there is only one man who is capable of safe guarding the souls of the West - Paschal I becomes the Patriarch of Rome (ushering in the Second era of Dual Papacy).
880AD: Byzantium falls to the HRE – Rome restored as the sole papacy (Holy Roman Empire borders: the Seine and Rhone rivers to the west, the Order and Danube to the north, and Anatolian coast to the east).
1060AD: HRE grants aide to the Iberian kingdoms in their war against the Moors.
1130AD: Civil war in the HRE. Rome sacked and occupied – Pope Innocent I moves the Christian capital to Ravenna seeking the direct protection of the Holy Roman Emperor Conrad II.
1150AD: Civil wars are concluded in the HRE, Henry IV – Conrad II brother, is crowned the new Emperor. Rome resorted as the papal seat.
1155AD: Civil war once again in the HRE. Bishop Nicolas Breakspear, a papal legate and popular among populous and the clergy of northern Gaul, the Brittan Kingdoms, and Norway, proclaims himself Pope after the death of his mentor Pope Eugene III. Breakspear’s radical ideas about reforming what he saw as a corrupt Church had made him many enemies in Rome as well as among the HRE Nobles. At the onset of a fresh civil war and current pretender to the throne using the Faith as his reason for regicide and usurpation Bishop Breakspear (residing in Arles) proclaims himself pope and is supported by a large following. Enter in the dual papacy of Arles and Rome.
1181AD: Arles established as the sole papal seat. Civil wars in HRE end – HRE breaks up into several warring kingdoms (the Kingdom of Germany by far the largest and strongest). With relative peace established and many new leaders striving for validation among their peoples a call goes out for Rome to be established as its own state and the Papacy restored to the eternal city. Arles tenuous grasp as the center of the Christianity (born of war and strife) finds its support slipping in the face of peace.
1182AD: Pope Adrian III, a reformer much like Bishop Breakspear who founded the papacy in Arles, had long felt that the corruption and warring nature of the West was no place for the Papal Authority. He felt a uniting force was necessary to purge the sins of the west and unite the land in a new brotherhood. He found support among many of the fledgling rulers especially that of Philip II King of the Franks, and the Iberian nobles who had been under constant war with the Muslims for more than a hundred years. He preached for a holy crusade to free the ancient lands from the horrors of heathen rulers.
1185AD: The First Crusade.
1186AD: Saladin’s forces were able to inflict many defeats on the invaders but were unable to win the battle for Jerusalem. In retreat and facing rebellion in Syria Saladin allowed the crusaders to entrench and establish the Kingdom of Jerusalem. Pope Adrian III wasted on time in establishing Jerusalem as the center of Christian authority – and forever separating the church from the burden of dealing with constant western warring.
1191AD: Pope Adrian III successor Alexander IV would not have long to reside in Jerusalem as Saladin’s forces renew the war.
1193AD: Jerusalem is retaken. Alexander IV flees to Alexandria (his native city) establishing that as the new Papal seat – claiming Alexandria to be “the guardian of the west”. He also places many privileges and special praise on Byzantium as sharing the burden of the gateway to the west (the Byzantine Empire, which extends through Greece, Macedonia, and the Anatolian coast).