No. HRE should stay HRE. It was a silly attempt at usurping Roman glory, but it's historical. It simply happened. No point in denying its existence. It will also spice things up in the game, just as it did in history.
You people will really say anything just to never admit it was a Roman state
It was
"a continuation of one of the regional Roman governments but not the continuation of the Roman state"? Oh, come on. ERE was not some client state formed from locals. In the IV century it was as Roman as Sicily or Galia Cisalpina. The only problem you have with it is that it outlasted the WRE. Really. If both ERE and WRE survived somehow to the 1337 together, with WRE still owning Rome (even if just it and its vincinity) - there would be no problem with the "legitimacy of ERE". But just because Rome fell out of Roman sphere of influence - you all suddenly start having problems with it. As if it really changed anything for the functioning of the eastern Roman state.
It was a huge empire spanning from Portugal to Egypt. It's not surprising that, after 1000 years of turmoil and mass migrations, only parts of it will remain. Hell, even that's pretty impressive that in 1337 part of it was still on the map, after all that had happened in the meantime.
"Different culture and language"
The Roman Empire at its height was very multicultural. It was not an ethnostate. In the IIIrd and IVth century even senators and emperors hailed from numerous provinces. Emperor Maximinus Thrax (235-238) was literally "from Thrace". Do you doubt his "Romanness"?
Roman east was always dominated by hellenic culture and language since the Roman foot stepped there for the first time. For centuries masses were using Greek language and local documents were written in Greek. And yet no one doubts the province of Achaia or Macedonia was Roman under Hadrian or Commodus. Just because centuries later Latin was replaced by Greek ("Romeika" - "Roman" language as they called it) in official documents/court changed very little. It simply acknowledged the fact that majority of citizens used that language, so it's easier to use it in administration.
"Different politics"
What exactly was different? What can't be explained as simple evolution of the same polity?
Roman kingdom of Tarquinius was different from the Roman Republic of the first Punic war, which was different from the Roman Republic of Julius Caesar which was different from Roman Empire of Nero which was different from Roman Empire of Constantine. And yet they all are perfectly "Roman", right?
Eastern Roman Empire is just a next step in this absolutely natural evolution and it continues uninterrupted up to the 1453.
Besides, I don't think it's up to discussion anymore really, as devs rightfully openly describe ERE as "the Roman empire" (there won't be a way to "restore the Roman empire" as "Byzantines" because it doesn't make sense, as "Byzantine empire"
is the Roman empire). It's not a matter of "whether it was Roman or not" - it's only the matter of "which name would be best for this Roman empire" now.
"Legal system"
Now that's strange. "Byzantine Law" is basically the most Roman law in the world still in existence in 1337. Corpus Iuris Civilis and all that. There will even be an event in EUV about that as in 1345 Roman judge from Thessaloniki created a handy compilation of it in only six volumes, the
Hexabiblos.