I like the term "Byzantine Empire" specifically because it sounds cool.
However, I maintain there is not some point in time where you can logically "de-Romify" the Eastern Romans based on some event/trend/factor. As I have posted before on this topic, being "Roman" by the Fifth Century AD was no longer AT ALL descriptive of Italic ethnicity, speaking Latin or being from the city of Rome (or Italy). It just wasn't. People had begun to view Rome as the "one true empire of God on Earth," and being
Roman was entirely dependent on fulfilling that vision and carrying on that legacy. If the US split into the Western US and the Eastern US, and the Eastern US (i.e., where the original English Americans started out "civilization") fell, NOBODY would consider the Western Americans as not just "Americans," even if they ended up speaking Spanish in this future scenario.
I like the term "Byzantine Empire" simply to help people unfamiliar with history distinguish Antiquity from the Middle Ages, and again because I think it sounds cool.
However, if you are trying to pinpoint some time in which the Byzantines are no longer "Romans," I think you are COMPLETELY missing the mark. I highly encourage all to watch this video and many of the other ones from this channel:
Were the Byzantines Roman?