The theory is that Brennus is not actually a name, but rather a title to be given to chieftains or warlords of high status, but not necessarily a king. Brennus, therefore, would probably make sense as a title to be spread rather than the personal name of this celt, much like how Greeks and Romans often categorized peoples by a political title rather than the tribe's personal names (see the theory on the Huns as actually being a title, rather than a ethnic designation) due to linguistic differences. The barbarians themselves used to do the same thing, apparently only knowing the emperor as "Caesar", even if that wasn't his actual name.

Your'e not wrong about the Roman historians tendency to color their earliest histories with allusions to other events in Greek history, so it's understandable for the concern. In defense of Livy, there seems to be indications that the Greeks themselves had recorded the sacking of Rome by the Celts, at least according to Plutarch (Camillus 22.3-4) and Pliny (NH 3.57), who argue that authors and thinkers of the 4th century BC like Aristotle and Theopompus wrote about it. This isn't surprising, since the intrusion of the Senones and ravaging up and down Italy until they eventually reached Syracuse and Sicily to serve under the tyrant Dionysius I would have sent shockwaves to the greek speaking peoples of Magna Graecia and Etruria (the Etruscans). I imagine that the Romans themselves would have seen as a big deal considering they were one of the premier Italic/Latin city states at this point.

In addition, I am almost certain that the Romans would have taken special care to remember the name/title of the only person to have ever sacked the city up to that point, even if they may have not understood the exact context of that name.