Many Western polities have sought the title of the Roman Empire and the legitimacy it endows. Constantine split the empire, and Constantius II did not rule over Rome (or many other parts of the empire). Is it even reasonable to assert it was still the Roman Empire after the fracture?

Is it legitimate for the Eastern Roman Empire to claim the legacy? I think so, and I think they have among the best arguments for it. Conversely it is also legitimate to note the major differences between the two and the fact that discontinuities do exist.

"Capital of the Western World for a Cool Thousand Years" (TM)

They actually called themselves Ῥωμαῖοι, not Romani, and not every description needs to be emic. There's nothing wrong with the label "Byzantine" except to Byzaboos.