Tullianum is Cicero, Marcus Tullius Cicero; earlier ages tended to call him Tully, where we call him Cicero. That's probably not what's troubling you with the passage.
Tullianam is not Cicero. Tullius (Cicero) is, as I noted in my post above. Tullianam is taking the name Tullius and turning into an adjective: Tullianam facundiam - eloquence on the level of someone like Tullius. "The eloquence of Tullius" would, rather, be written as "Facundiam Tulli(i)"
Anyway, I had class today, so I mostly got my question answered:
cui scribendae atque explicandae non meum ingeniolum, quod exile et paruum, immo poene nullum est, sed Tullianam par erat desudare facundiam.
Basically the answer to the problem I had is that Tullianam facundiam and meum ingeniolum are accusative subjects of the subordinate clause kicked off by "par erat" (it is suitable/fitting). An example of this happening elsewhere in Latin would be Plautus's Poenulus: "liberos homines per urbem modico magis par est gradu ire" (It is more fitting for free men to go through the city at a moderate pace)
Arranged in a less confusing way, the sentence reads:
cui scribendae atque explicandae [vitae Karoli Magni], par erat non meum ingeniolum, quod exile et paruum, immo poene nullum est, desudare, sed Tullianam facundiam desudare.
"For writing and explaining this thing, it was sufficient, not that my small genius be made to sweat, genius which is minuscule and tiny and, indeed, scarcely exists at all, but that it be a Tullian eloquence which be made to do so."
Also, Latin scholarship was crap during the Middle Ages. People serious about the language have to learn Medieval Latin as its own whole thing. So don't assume this:
means anything.
Post-Roman Latin competency varies quite wildly. However the Latin of the 9th century, in which on the one hand you have West German and French writers for whom Latin was, quite literally, their native tongue, and on the other you have Irish, British, and Saxon writers who were exceptionally well-versed in (and in fact wholly and singularly taught from) Classical Latin, was generally pretty good; for the most part the Latin of that period would be quite on par with most of the Classical Latin you would have seen 6 or 7 centuries prior. It wasn't called the Carolingian Renaissance for nothing. It's not until the 1000s and beyond that you start to see some really funky stuff happening with Latin, and even then, it varied. The Latin of Peter of Abelard, for instance, is quite the equal to just about any Classical text you might find.
Einhard, in particular, was an
exceptionally talented Latin writer. And more to the point, Vita Karoli Magni was a book designed in large part as a project for him to show off just how exceptional his Latin prose was. Any attempt to make your way through his Prologus makes that rather immediately apparent.