It's a bit weird to write that the Romans of Constantinople "claimed" to be the "heirs" of the empire; as far as they were concerned they simply were Romans and their empire was obviously the same empire as Augustus', albeit with a change of capital city and so on. And given that they were Romans and the Emperor of Rome did become the Emperor of Constantinople, you have to do some work to argue against them.
In particular, very unsurprising that they'd share continue to like the same colours of shoes.
It's not hard at all to refute the Byzantines' claims to being the Roman empire. If your empire doesn't include Rome, it cannot be the Roman empire. Similarly, Taiwan is not China even though they are the continuation of the old Chinese government.
When the bizantine re-took part of Italy, the local germanic tribes who had occupied Italy identified them as Romans, which is why to this day there's a geographical region called "Romagna" a few hundred kilometers from the city of Rome.
That's silly. The Roman empire could not have existed at all, delusional people claimed roman citizenship but lived far beyond the city limits. You can't be a citizen of a place you've never been, so the whole thing is just dumb and a myth.
I can't believe you buy into this absurd nonsense that there could even be a Roman Empire without a megalopolis covering much of Europe, North Africa and the Levant.
In particular, very unsurprising that they'd share continue to like the same colours of shoes.