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."