The distinction between "Merovingian" and "Carolingian" Frankish names is essentially a later historiographic convention, appearing first when history books were directly written in French, around the Late Middle-Ages but still borrowing from Latin sources on which the transliteration of Hlodovec was written "Clovodicus" and then "Clovis" (C standing for the sound [x] inexistant in Latin or Romance languages) whereas the name was transmitted in spoken language without the formal C, giving Louis (or the more latinist "Ludovic") but also on the same model "Lothaire" from Hltothar -> Clotharius.

Essentially, Clovis is the archaising Medieval Latin of Hlodovek, whereas Louis is its evolution in Old French. How much the distinction was lost or not to late medieval writers (such as the Grand Chronicles of France's) is something I wouldn't be able to answer but the popularisation of the distinction comes from this litteral vulgarization of history.
Even, though, if the distinction wouldn't have been perceptible in the early IXth century (especially as the use of early Merovingian names might have been the whole point) it have to do with how Carolingian chroniclers portrayed the dynasty they took the power from and which ambivalent legacy it had in medieval royal historiography.

The "coup" of early Carolingians, for all the effective strength it drew from their aristocratic networks and the alliance with the Church, still had to be justified a posteriori not to appear as mere "usurpers" regardless how justified it could be, especially as Merovingian kept some prestige in the eyes of VIIIth century Franks but also a significant judicial and political power (much more so than was accounted for later) : for a time, whie their father was able thanks to his own personal prestige to be the majordomo of no king, Peppin and Carloman had to reset a Merovingian on the throne to legitimize their own rule for a brief time.
They then had to be displayed as "rois fainéants" (do-nothing kings) as they were know in later French historiography : not good too anything much safe drinking, eating, sitting on their backside all day and never, ever, allowed to do any political action they wouldn't be able to do sanely; something the much more able Carolingians were more than able to do.
Unsurprisingly, the main promoters of this idea are to be found in Carolingian entourage. The continuators of the Chronicle of Fredegar (originally stopped in 641) were first likely a clergyman dubbed as "the Monk of Laon", probably associated with Austrasian aristocracy, and then the half-brother of Charles Martel, Childebrand, hailed the nobility and heroism of his famile while describing Merovingians as depraved imbeciles. One of the main promoters of the notion, and certainly the most famous, was Eginhard. Taking over the "useless king" trope, he even insisted on their decidely un-royal behaviour by stressing that, they were transported on chariot driven by oxen, just like "rustic people" did, the height of horror for Carolingian and the rest of the militarized Frankish aristocracy having adopted the horse at its main warfare and social apparatus since at least decades. And, on closer observation, what the whole thing about long hairs? Didn't it reeks of paganism?
Of course, this extremely tendencious take on Merovingian is quite remote from what we know about them (and in some part, how they appear at their apogee being more powerful than Carolingian kings at their height) : princes of a light-weighted Roman state in Gaul and overlord of peripheral principalties , administrating with great care the royal demesne and the public fisc, scholars able to write regularily in Latin, and so rooted down in Biblical kingship they adopted davidic display of royalty (including the long hairs and beard of the Old Testament. But, as the Merovingian dynasty disappeared (partly because of the result of violent dynastic "pruning out" not unlike what was practiced by Ottomans), it was easier to re-write their history in a way to both justify the Carolingian takeover and to have them displayed not as usurpers but as a dynasty chosen by God and thus less challengable.

But why, if this is the case, Charlemagne would have given his son a Merovingian name? And why this son would have given them to two of his sons? Even if Charlemagne quite possibly had Merovingian ancestry trough his mother, Berthe, it almost probably wasn't the case : such ancestry would have been far too contrary to what they tried to convey, especially as a good part of upper Frankish aristocracy was probably related to the former dynasty trough women.
Well, it was hard to entierely write off Merovingian out of Frankish history and historiography : the previous Frankish chronicles were still avaible and depicted rather powerful, resolute and pious kings; figures that were known to all. While they were potrayed at times in these chronicles in a negative light, as violent, forceful and with a limited sense of kinship; these were necessary vices of royal authority (or at least, vices that Carolingian practiced too) and less prone to discredit a dynasty than depravity, uselesness or the contempt they cause in their subjects.
Thus, Carolingians seem to have elected to consider the earliest Frankish kings (while fairly obscure in their own time, especially compared to their descendents) as "the good ones" : a bit rustic and rough around the edges, but for all their own prowess had their lineage promised to degeneracy. A fragment found in the continuation of Fredegar, describing the how Cilderic dreamed of "lions, rhinoceros and leopards", then "bears and wolves" and eventually "dogs and similar animals eating each other" and how his dreams were interpreted, quite biblically, by his wife prophetizing his son (Clovis) would have the bravery of the lion, and his own sons like the rhinoceros and the leopard, their own sons greedy and powerful as bears and wolves, and their sons eating each other like dogs barely different from their subjects, all destroying each other.
Coherence was safe : one could still point at how the desintegration of Merovingians was bound to happen and still consider their early ancestors as overall convenable people, that were indeed powerful, Christians, worthy creators of the Frankish realm Carolingian inherited from their own Christian values, as much as they inherited the imperium on Christians from emperors as Constantine or kings as David.

As such, it wasn't really contradicting having Carolingians using besides their own dynastic patronyms (e.g. Charles or Peppin), the names of the earliest Christian Frankish kings : doing so was stressing a political identity setting them apart from the imperial aristocracy they weren't that socially or politically distinct from. In the course of desintegration of Carolingian Francia, these referrence were even more stressed on : Charles II was sacred in Rheims by Himcar, a bishop claiming having stewardship of the annointed oil miraculously provided for the baptism of Clovis and thus, bringing Himcar's bishopry a strong prestige while the details of Clovis' baptism are actually mostly lost to History ( u/Libertat). In mixing sacre of Frankish kings (not practiced by Merovingians, but initiated by Carolingians) and a Merovingian political and religious "ancestry", this action perpetuated the ambiguity legacy of the "first race of kings" in royal historiography and hagiography not just under Carolingians but under Capetian as well (even if they did not claimed Merovingian ascendency nearly as much they did so with Carolingians). Frankish kings being called Louis or Lothar weren't as much claiming a familial ascendency, than a royal legitimacy and legacy.