How might a split between the Eastern Orthodox Church and the Roman Catholic Church affect the future of the Byzantine empire?

The Byzantine Empire, also called the Eastern Roman Empire, was the half of the Roman Empire that did not fall in the 400s. Until recent decades, Western observers and historians paid little attention to the Byzantine Empire because they deemed it to be Eastern, exotic, and, therefore, backward. In fact, the Byzantine Empire was very advanced for most of its existence, with major contributions to world civilization, such as Justinian's Law Code (529–34), and was not as exotic and foreign as Westerners tended to think.
Having undergone a Christian conversion experience at the Battle of Milvian Bridge (312), Emperor Constantine the Great issued an edict of religious toleration, called the Edict of Milan (312), and later founded Constantinople (324) on the site of the existing city of Byzantium, where he became patron of the Church and called the First Ecumenical Council at Nicaea (325). The Byzantine Empire, a term used by historians to differentiate between the Roman Empire of old and the eastern half based in Constantinople after the fall of Rome (476), was a continuation of the Roman Empire, with major Latin elements, but it also assimilated many Greek elements by virtue of its geographical location. Meanwhile, the Church in Rome fell under the onslaught of Germanic tribes and underwent its own transformation, no longer sponsored by the imperial state, grappling for traction under various Germanic kingdoms. As Rome began to regain its freedom, Constantinople had fashioned itself as the New Rome. To make a long story short, by 1054, when the schism between what subsequently came to be known as Roman Catholicism and Eastern Orthodoxy occurred, there were substantial differences between the Christian West and the Christian East. Tensions arose over 1) the position of the pope within the Christian Church and 2) the filioque. Rome believed that the pope's position as "first among equals" translated to power, whereas Constantinople believed that the title was largely honorary and that the pope had to act in a conciliar fashion, in consultation with the other major patriarchates, which included Constantinople, Alexandria, Antioch, and Jerusalem. Rome had also added the phrase "and the Son" (referring to the procession of the Holy Spirit), known as the filioque, to the Nicene Constantinopolitan Creed. The papacy claimed it was adding something that was always implied but unspoken, whereas Constantinople perceived the addition to be a corruption of Christian theology. These differences led to mutual anathematization of Rome and Constantinople in 1054.
What long-term impacts did 1054 have on the Byzantine Empire? The list is long, but the most dramatic outcome was the hostility of Western Christendom, which came to a head in the sacking of Constantinople by the Crusaders, namely the Franks and Venetians, in 1201–1204 as part of what came to be known as the Fourth Crusade. The city was burned and sacked, with large-scale slaughter, looting, and the destruction of sacred relics. Subsequently, Constantinople was forcibly subordinated to the papacy under the Latin Empire for over five decades. By the time Constantinople was liberated in 1261 by the Empire of Nicaea and returned to Orthodoxy, the Byzantine Empire was a shell of its formal self, resembling a weak feudal kingdom more than a centralized imperial state. Having never recovered its former glory, the empire gave up more and more ground to Islamic invaders, with the Ottomans under Mehmet the Conqueror striking the coup de grace, or final blow, with the sacking and seizure of Constantinople (1453).
For more detail on the topic, might refer to a general Byzantine history text, such as Timothy E. Gregory's A History of Byzantium.