It's also worth noting that "Byzantium" and "Byzantine" show up quite frequently in our classicizing Byzantine histories (although it's unlikely they were being thrown around in demotic speech), but just in relation to Constantinople and its inhabitants, never Romania or the Romans more generally; Attaleiates exemplifies this when he notes, of Constantine IX in the mid-11th century, that "he brought strange animals from foreign lands, among them an elephant, greatest of all four-legged creatures and an amazing sight to the Byzantines and all other Romans who happened to see him as he paraded by." Roman coins were, as you mentioned, minted almost exclusively in Constantinople (that is, Byzantium) in the Middle Byzantine period, so it wouldn't be illogical, in that capacity, to call them "Bezants", completely aside from the issue of Roman identity and its recognition (or lack thereof).
Everybody liked to call themselves Roman,...... the Germans, the Russians, the Seljuk Turks, and even the Hellenes. It was a prestigious thing to do!
The truth of the matter however, is that if there are some people in today's world that have the right to claim ancestry from the old Romans in the tradition of Julius Caesar, Pompey, Antony, and Octavian or Cicero, as well as of Marius and Sylla and of Scipio, etc.,..... are the Italians.
I know a lot of barbarians entered the Italian peninsula after the fall of the Western Roman Empire (WRE), but those people they intermingled with the local Roman inhabitants and eventually, they were absorbed or assimilated by the pre-existed Roman population.
Something similar happened in the Byzantine Empire (BE), where originally when Constantine the Great moved the capital of the Roman Empire (RE) to Constantinople, the governing elite Roman/Romanized that he brought with him they ruled over a Hellenic/Hellenized population and this situation lasted for almost 250 years. This lasted until the last quarter of the 6th century, because by that time even the ruling elite had become and was of Hellenic/Hellenized extraction, as the overwhelming majority of the population had been from the very beginning of the BE!
Nowadays we have people that out of wishful thinking, or something else they like to call the BE as being Roman.
The renaissance historians were very ingenious in calling it Byzantine Empire because they realized there were important and inherent differences with the old RE.
One does not become Roman by simply calling himself as such, like the Germans, the Russians, and the Seljuk Turks eventually realized; and that in the same manner that the Greeks/Hellenes of the BE eventually finally also realized, and not only this .......but they became very proud of their Hellenic ancestry of ancient times!!!