Considering the Byzantines thought themselves Romans, how did they take the news there's a new Rome in town?
Not well, though pretty much how you'd expect a Roman to react.
From luitprand of Cremona, an ottonian ambassador:
But, to increase my calamities, on the day of the Assumption of the Virgin Mary the holy mother of God (August 15), there came-an evil augury for me-envoys of the apostolic and universal pope John, through whom he asked Nicephorus, the emperor of the Greeks "to close an alliance and firm friendship with his beloved and spiritual son Otto "august emperor of the Romans." Before the question as to why- this word, this manner of address, sinful and bold in the eyes of the Greeks, did not cost its bearer his life-why he was not annihilated before it was read, I, who, in other respects, have often shown myself enough of a preacher and with words enough at my command, seem dumb as a fish! The Greeks inveighed against the sea, cursed the waves, and wondered exceedingly how they had been able to transport such an iniquity end why the yawning deep had not swallowed up the ship. "Was it not unpardonable," they said, "to have called the universal emperor of the Romans, the august, great, only Nicephorus: "of the Greeks"';-a barbarian, a pauper: of the Romans'? Oh sky! Oh earth! Oh sea! But what, "they said, "shall we do to those scoundrels, those criminals?" They are paupers, and if we kill them we pollute our hands with vile blood; they are ragged, they are slaves, they are peasants; if we beat them we disgrace not them, but ourselves; for they are not worthy of the gilded Roman flail and of such punishments."
Did the Romans even refer to themselves as being apart of a "holy" Roman Empire? Seems a bit weird then again I'm not a historian.