You miss the point though that none of these states (HRE, Ottomans, Franks, Sultanate of Rum) used Roman law (Corpus Juris Civilis) they used either their traditional barbarian tribal laws or islamic law. So firstly a completely different political system.
So there is no legal continuation
The HRE, Ottomans, Franks & Sultanate of Rum had no claim to the office of Emperor. The imperial insignia in Rome was sent back to Constantinople by the Goths.
So there is no official transfer of the Office of Roman Emperor to any of those states. One can't just call themself a Roman Emperor any more than I can call myself President of the USA without going through the American political system
The Roman Emperor followed in the traditions & political philosophies of the Roman state laid down by Augustus, Marcus Aurelius and other emperors. They also shared responsibilities with the Senate. There is no such comparison in any of the other states. To be a Roman Emperor was to be more than an autocrat - the very concept of Rome and being a Roman was a political ideology.
The political ideologies of the Roman Emperors have nothing in common with the HRE, Ottomans, Franks & Sultanate of Rum (these were states divided purely by race and language). The Roman Empire had evolved beyond that (hence why you see many Armenians, Illyrians, Isaurians etc becoming Emperor, not just Greeks)
As for the HRE, Ottomans, Franks & Sultanate of Rum there claims to be Roman Emperors was purely symbolic or an ego-trip. I can call myself God, it doesn't make it true.