The thing is - it's simply not historical. The Byzantines never meddled in who got to bethe greatest among equals.
What?
Basil's third son Stephen was made patriarch when his brother became emperor though we are told he had no religious experience and was fond of horses and women. The whole spiel with Ignatius vs. Photius? The Iconoclast issue? The patriarch of Constantinople was more of political than religious role and emperors had tremendous power in choosing who took office.
Patriarch Nicholas Mysticus?
I admit, there was a limit on their power, but the patriarch was appointed by the emperor who himself represented the will of the divine on earth and was seen as the conduit through which the beneficence of God may be bestowed upon all of his children.