Hello from France,
The only problem with Charlemagne's sister Berthe, his brother-in-law Milon and his nephew Berthelot, is that they're only mentioned in legends, and several centuries after Charlemagne's time. Berthe and Milon appear first in a text from around 1200, at best. Berthelot (more exactly "Bertolai" in the first version) appears in a chanson de geste from the 12th Century, "Renaud de Montauban" (and it's unclear if he's supposed to be the son of Berthe, or of some other sister/brother of Charlemagne). So more likely at some point someone in the Berthelot family read/heard those legends and thought "oh, that's fun, there are people called Berthe and Berthelot in there, what if we were their descendants? That would make us related to Charlemagne."
Regards
Vincent