I realize that scholars have many different opinions on this, but I am of the persuasion that the Byzantine Empire, and Holy Roman Empire (not to mention other Rome wannabes like Russia) are different enough from Ancient Rome, that I really do not consider them continuations of the Roman Empire. You really can't have a Roman Empire if you don't have Rome. If you disagree, I'm fine with that.
The HRE was very different.
The BE was just the Roman Empire, centred in the capital established by the Roman Emperor Constantine, comprising many core parts of the Roman Empire. No one living at that time would have seen it as anything other than the Roman Empire.
This is what it looked like under Justinian
Anyway, regardless of what later historians came to call it, the problem it poses for the idea that Christianity caused the end of the Roman Empire is that the Eastern half, whatever you want to call it, went from strength to strength and only the Western part was made effeminate and weak or whatever mechanism was supposed to be at play (even though many of those who overthrew rule in the West were themselves Christians).
It's a view steeped in prejudice (particularly anti-Catholic), Western European chauvinism and poor history given there are numerous far more plausible and direct reasons. I'm not saying that you hold these views, but that is what drove the popularisation of such ideas.
Western-Eurocentric views of history are very common, and I think we are better off without them and can understand the world better without them.
For example unless we see Rome as specifically a "European" Empire rather than the North-Eastern Mediterranean one it really is, the Roman/Byzantine distinction makes less sense and seeing the Graeco-Roman world as specifically the foundation of Western civilisation makes far less sense also. It is basically Germanics appropriating classical culture to cover up their parvenu roots (and often to bolster racist theories as if white folk are indeed genetically superior, they had to have been superior in the past too rather than simply "barbarians").