1. It is not obvious that the Byzantine Empire has more legitimacy than the HRE. After some time arguing my points for the primacy of the Papacy in these fine matters and of the precedence set by the translatio imperii, and have not yet once been convinced of the primacy of the Eastern Emperor in defining titles of Rome or seen evidences that without doubt compel me to say Byzantium is above the HRE or for that matter France, in maintenance of the Roman Empire. Just as there are arguments to de-legitimize the HRE or France, there are arguments to de-legitimize the Byzantine Empire, even prior to Islam.
So do not grand-stand upon some sort of victory, that the Byzantines are by virtue of common wisdom more legitimate. It is not so, and never will be so.
2. I would certainly argue for the Byzantine empire having no longer remained the sole Empire of Rome after the Iconoclast Crisis at least and to a degree prior. Translatio Imperri permitted the removal of any title by the Papacy, as supported by the entirety of the pentarchy at the time and the primate of Rome. Surely,afterward however, Byzantium once more was recognized by other as an Empire of Rome, but its separation after 1054 and then inflamed by disagreements in the 12th century led to its conception of Rome as disappearing. Rome and the Roman Empire in the Latin world became certainly a possession of the Papacy who ruled Europe as a federated empire of feudal vassals in a sense little different from the Zhou Dynasty of China or the Ashikaga Shogunate. Over time, disputes over the course of this federated empire and to whom authority over Europe truly rested led to the eventual decline of Papal lordship in Europe first in 1305 and then accelerated to huge degrees in 1379, 1414, 1419, etc... and afterward, only a semblance of this system remained and we could argue was breached with the English Acts of Supremacy.
Thus, one might see the point that I make, after the 8th century crisis, the Papacy revoked Empire and took upon itself the Empire in the Latin world and distributed this glare unto others. The famed celestial body allegory of Innocent III, 'one light is made evident by the divine and this light is the sun (Papacy) who in turn reflects the light illuminated by the divine unto all celestial bodies (the lords of Europe/namely the Germanic/Latin lords)' a most sublime description of feudalism and of the custom up to his time of the nature of European society after the fall of the Roman Empire and its transmission through transference unto the Latin west.
So, the Byzantine Empire ceased to continue to be the Empire of Rome after the Iconoclast Crisis and the Translatio Imperii. However, after reconciliation, there became two mutually affirming empires of Rome until friction over linguistic issues, cultural differences and the decline of Byzantine political authority (defeated in wars) led to the vast majority of Latins becoming forgetful of the Eastern Empire and often disgusted by both her weakness in the face of Islam and apparent haughtiness.
