Thing is, Charlemagne wasn't a Roman emperor, regardless how holy : his use of the imperial title tended to be non-descript ("Emperor, King of Franks") or convoluted (such as "Charles, most serene Augustus crowned by God, the great, peaceful emperor ruling the Roman Empire") and the succession plan of 806 doesn't account for the imperial title's transmission to his main heir; probably because it was a personal title and not supposed to be inherited initially and whom transmissionl owes probably much to the fact only Louis survived, out of the planned successors.

The Roman Empire had an ambiguous reputation among Franks in the VIIIth and IXth centuries : while it was clearly seen as a powerful state with a lasting legacy and whom works deserved to be preserved, it also suffered from the unpopularity of Constantinople's (people largely identifying the Eastern Roman Empire as...well, the Roman Empire) in religious and political matters, but also from the historiographical projection of a Frankish identity rooted down in Christianity and replacement of the Romans in Europe, both as a state and a people; especially in the light of the unpopularity of Eastern Romans (mostly due to religious reasons).Charlemagne calling himself "Roman" would have been an important renouncement to his identity as a Frankish and Lombard kings (both titles he claimed and displayed until his death, together or even at the exclusion of his imperial title).

Carolingian authors themselves didn't often used "Romans" as referring to the ancient or medieval people, but rather to the city itself and their head, the pope. Even religious authors, which were the more prone to draw a parallel and a continuity between the ancient Romans and Franks, took a great care distinguishing both (for instance Ermold the Black wrote about a Maur mocking Franks attempting to take a Spanish city, pointing that Franks couldn't take the strong walls built by Romans of old).

The empire Charlemagne ruled was Roman, because it was a Christian empire sanctioned by the spiritual leader of Christianity, the Roman pontiff. And this is what Charlemagne ultimately claimed : an imperium over Christians, either directly on his dominion, or as a "Christian commonwealth" over minor peoples he would still dominate as he dominated the Church the same way Late Roman and East Roman emperors did for centuries. This power over religion and religious political matters is essentially the only sphere an actual imperial policy can be seen, the rest of Charlemagne's reign being largely in line, both ideologically and practically, with his predecessors'.

It doesn't mean that no connection was made, especially after Charlemagne with the monopoly of clerical circles on intellectual culture : antiquated depictions inspired by ancient Romans features were present (such as the famous denarius depicting Charlemagne as emperor with a laurel crown; but as well the illustration of Frankish warriors dressed with what Franks thought Roman armours looked like), but they remained uncommon and mostly from religious circles.
Ermold le Noir describes something that point well what was at stake in the decoration of the aula regia of Aachen during the reign of Louis, with the figures of David, Salomon, Romulus, Augustus and Constantine : Roman emperors are set in the continuity of Hebrew kings (which were a strong model in Frankish kingship since the VIth century), a translation of rightful political power that came to the Frankish emperor.

As such the intent behind what was originally a personal title, the crowning of Charlemagne's "career", was not to claim Romanity as did Eastern Romans emperors, far from it he stressed his "Frankishness" in his imperial displays, but the imperial power itself, backed by religious legitimacy.