There was no legal successor for the simple reason that there was no clear law to establish Charlemagne's theoretical succession. This was before proper feudal succession laws, when heirs were established either by the man in power as he was still alive and regnant, or at his death.
We can retroactively apply later succession traditions/laws, and if a ruler gathered enough power in the time it's probably how they would justify being the heir of Charlemagne too. Let's keep in mind that in western Europe, the key concept at play there wasn't that there was a sacred imperial bloodline (that's more of a fantasy trope), it was the
translatio imperii, that is a linear transmission of imperial power from the Romans to the Franks and then the Saxons (HRE line), but there wasn't just one interpretation. For other others, it was Greece to Rome to the Franks/France, for instance.
You can find
a few other examples on the wikipedia page.
That's why there's two decisions in game. One is somewhat historical with the foundation of the HRE, modelled after the Saxon
translatio imperii.
The other represents a superior grade where a ruler managed to unite the german, italian and french parts of the Carolingian/Roman empire.
As a side note, claiming to be a direct heir of Charlemagne could help to claim his empire, for sure. However, the real claim was one of power and organization of the empire. When Otto rose to power, he beat the barbarians/pagans (the Magyars), was crowned by the Pope and established the imperial church system. If I had to imagine factual prerequisites for a ruler to claim Charlemagne's heritage, it would probably be those three things (that plus ruling a large territory in western Europe, of course).