Lothair intended to inherit the whole of the Carolingian/Holy Roman Empire for himself, while Charles and Louis intended to divide the empire.
It's not really was at stake, as
@Thoresby said but to enforce a share of Carolingia that would benefit him as emperor and (technical) overlord of the Carolingian ensemble. The principle of division of the state was firmly entranched at this point since Late Antiquity in western Romania (altough it had different forms), the main change for what matter Carolingians being the appearance of a feudal order, as in privatization of public power (long story short, it comes from Peppinid coup and takeover of kingship) which pushed the division into a greater scope.
But what if Lothair had managed to defeat Charles and Louis, ascend to the imperial throne, keep the empire united, and estabilish a system of primogeniture?
Crusader Kings 2 isn't really a good model for what matter succession laws, no matter how much it get brang on the table : the idea of distinct successions system doesn't really exist before the Late Middle Ages, and what rules before are tradition, remembered precedents, and political situation : would Lothar I wins the Battle of Fontenoy, his sons would probably requires their own share and would probably be supported by their uncles.
According to Peter H. Wilson the Battle of Fontenoy, because of how bloody it was, shocked the elite and they pressured Lothar to accept to "divide" the Empire into sub-Kingdoms between the brothers, Lothar was allowed first choice was choose the center of the Empire, anchored in Aachen, and Italy, anchored in Rome.
While not untrue, I think you can bring the opposition of Carolingian feudal aristocracy on an earlier date : the
"Field of Lies", notably, hints how much Franks favoured a settled peace against a re-edition of VIIth century
faida among Merovingians
. Now, I don't disagree with your point or your conclusions, but I think that's a tendency that comes futher than just the battle, from a Carolingian nobility that demonstrated its absence of will and enthusiasm to go kamikaze for their kings.
So even if Lothar wins the battle, the nobility may force him to accept a "division" between with his brothers and even his victory doesn't means that "Primogeniture" would become the norm, at this point tradition was what decided the succession and the Frankish tradition was to divide between the sons, with the eldest getting the richest parts of the land.
I'm not so sure, tough, that you'd end up with something this hugely similar to what happened IOTL : a big effect of 842 was that the interwebed loyalties among the three sons of Louis the Pious were more cutted down, with each king (technically) ruling over "his" aristocratic network. How it played for Peppin II after Verdun, Aquitain nobility being more divided and shifting between him and Charles, highlight the mess it could form.
I could see the emergence of a carolingian "dynastic system" as it appeared in the mid-Xth century meaning the various kingdoms of western Europe slowly emerging from Carolingia and their immediate neighbours, being ruled by Carolingians or "half-Carolingians" (as Athelstan)
different kingdoms ruled by Carolingians or Half-Carolingians (as some Wessex kings could be considered and probably were) forming a metapolitical ensemble, that would be certainly unstable and vulnerable (especially with the undergoing crisis, economically, politically, militarilly and climatically respects) but could last trough the Xth and XIth centuries, as a softer version of what happened IOTL, possibly preparing a different geopolitical outcome, with less of a "feudal revolution" :
I agree that the Carolingian Empire was a weak empire to begin with that only the successors of Charlemagne found worth maintaining for practical reasons and that it's quite unlikely to see it last even with a more confederal form (akin to how
@RGB described Kievan Rus' IRRC, but probably on a more sophisticated scale).
Still, it would bring significant changes for the period, altough I agree it's not the result of a Lotharingian victory at Fontenoy only, but a Lotharingian dominance (politically and strategically) consequences.
You may have Luis the German ending as King of Bavaria instead of King of East Frankia, and with Charles's as King of Aquitaine instead of West Frankia, Lothar would still keep Italy and the rich valleys along the Rhine and the Rhone until the North Sea and the rest of the lands, I think he may give his nephew Pepin a new Kingdom as reward for his support and for making Charles King of Aquitaine, that would be a bigger powerbase than he had OTL but if things go as per Frankish tradition it will end up being divided between his sons with the eldest getting Italy and the Imperial title.
It's quite unlikely IMO : IOTL Verdun agreement more or less consecrated the existing spheres of influence and nobiliar networks of a Frankish dynasty : while Peppin (as well as Bernard of Italy) had a territorial title (and not popular, as King of Aquitains) it was more a way to settle their authority compared to the "popular" kings that was Charlemagne (Kings of Franks and Lombards*).
Note that his successors were non-descript emperors ("August Emperor") or kings, at least how they styled themselves, even if they were still considered as Frankish kings (from the source of their political legitimacy) : it's quite interesting, on this regard, that the peoples associated with Lothar, Charles and Louis were called (IRRC), Lotharenses, Carlesenses and Ludovici(?), with the fiction of an equivalent to a regnum francorum lasting for some times, until Charles III.
*A title he constistantly used up to his death, contrary to Emperor ruling over the Roman Empire