Remember that the question is How did the Carandini family prove its descent from Charlemagne?. In other words, as OP said, I'm asking how the Carandini family proved their lineage to Frederick Barbarossa during the 1100s, long before anyone knew about genetic drift analysis.

This is a fundamentally different question from 'was Lee a descendant of Charlemagne?' or even 'were the Carandini's likely descendants of Charlemagne?'. This question is about medieval verification practice, not about genetics. I have removed all off-topic comments.

My obscure knowledge of obscure italian noble families can come in useful here!

This is the Carandini family tree, composed in 1304. It's currently in a vault somewhere held by the Banco Popolare foundation. I definitely never tried to steal it.

In case that's too difficult to read don't worry, it's probably made up; that happens a lot with Italian genealogies. The Medici were also supposedly descended from Charlemagne, while the Visconti would have us believe they were Romano-Celtic patricians living on the shores of Lake Maggiore since the aftermath of the Trojan War. Of course, it's nearly impossible do disprove these stories, as is disproving the Carandini's descent.

The earliest verifiable Carandini ancestor went by the name Something-Something de Risi (we don't know his first name) who was a German soldier of fortune who accompanied Barbarossa to Italy in the late 12th century. Mr. de Risi was probably a distinguished knight that claimed descent from Charlemagne, although we have no way of verifying if it's true. Barbarossa did chose to reward Mr. de Risi with a bit of hand-waving though, letting him stick the imperial eagle on his coat of arms. Certainly, with Milan nearly demolished and its citizens scattered all over Lombardy, Barbarossa wanted to repopulate the city with people loyal to him, and appeasing Risi's genealogical demands would certainly be a means to convince his knight to stay behind in Italy. We know for certain that a certain Paolo Antonio de Risi was born in Milan sometime in the late 12th century.

If Barbarossa wanted to build a base of support in Milan, he would have had to do a lot more than relocate a few knights and confer titles onto them. Paolo Antonio's son Cristoforo de Risi moved (or probably driven away) from Milan sometime after 1200 and settled in Bologna, which wasn't any more pro-imperial but local politics was more chaotic and allowed more people to slip through the cracks. Cristoforo is considered the founder of the Carandini dynasty. I'd be willing to assume "Carandini" derives from the given name "Cristoforo" but the academic consensus is out on the etymology.

The Carandini have a boring history; they pretty much attach themselves to the pro-imperial Este Family who carved out a Duchy for themselves spanning from Ferrara to Modena. In the very early fifteenth century the Este award them the fief of Sarzano, near Modena, and confer onto them the title of Marquis. I can only assume they also happily smiled and nodded whenever the "Descent from Charlemagne" story was brought up, as the Carandini keep the imperial eagle on their arms. In 1524 Emperor Charles V awards them the title Counts Palatine. The claim didn't seem to bother him either.

Members of the family held political and administrative offices in the Duchy of Modena right up to Italian Unification and continue to distinguish themselves in the present day. You can fairly easily bump into one of them in the hallways of one of the older law firms in Modena. I think the current heir to the title is a neuroscientist, while his father went into academia. One or two were members of parliament.

The family history was published in 1997 under the title, I Carandini: la storia e i documenti di una famiglia plurisecolare by Gianna Dotti Messori.

As a related question, after the Carandini family were allowed to use the coat of arms of the Holy Roman Emperor, how did the Holy Roman Emperor ensure that they did not impersonate him on foreign matters?