My version of the Charlemain story is a little different from yours as I tracked most of it through various history books myself. I don't necessarily believe that it is true but it makes a good story.
It also goes back to Charlemain and through him to his documented line to Petronius Probus Roman Consul in 322 AD and thus probably to Quintus Anicious Faustus Governor of Numidia in 197-201AD and his wife Sergii Paulla.
However the link I have to Charlemain is through his daughter Princess Bertha and is completely legitimate.
I will attempt to email you a copy to the address that you have on here.