What if Clovis, upon implementing his new succession laws at his death[1], rather than dividing the realm up between his sons, instituted a primogeniture succession and had his eldest son inherit all of his domains (maybe with his younger sons being given smaller, subservient lands to the eldest)?
[1] From what I read, before Clovis did a sort of gavelkind succession among his sons, it was more common for his people to pass authority to their uncles and brothers.
Okay, that's a bit different to begin with.
The division of the Merovingian kingdom in 511 was more or less unprecedented, from what we know, in western germanic usages. It's quite probable, that it can be traced to a mix of germanic inheritance where every son have a similar share AND to the late roman usage to have different rulers on a same entity (you can find it in different kingdoms, such as sub-kings in Visigothic Spain).
It should be pointed out that it never was considered as a division of the realm : the frankish imperium remained as much unified than WRE/ERE were.
It eventually went down to two factors : fiscality (each share being based on cities and fiscal revenues) and organisation of territory (for occupation and defenses purposes : for instance, Aquitaine was often divided in as much equal shares than you had kings).
The public land isn't divided in different entities, but remains as belonging to only one kingdom, being trusted to different rulers.
Normally, each king was king of the Franks, without other mention, and was normally to support his brothers and uncles : now that's for the theory, and as in Late Roman Era, infighting over monopolisation of kingship existed. It should be noted, though, that outright refuse of support, as Carloman did on Charlemagne, generally meant a weakening of legitimacy, critically when it comes to a foreign threat or a rebellion.
That said.
In spite of what Crusader Kings taught us, you can't switch from Gavelkind to Primogeniture because you feel like it (critically when these are only historiographical concepts, not formalized historically).
Clovis, as all Romano-German kings, was deeply under the Late Imperial model influence, meaning less primogeniture than both the highlight on the unity AND the necessity of political compromise to make the first be maintained.
By 511, I don't think you could have Clovis having a revelation, and trying to impose on a whole society a totally alien succession law that would have only the merit to make historians of the nationalist era more comfortable about their national historiography.
It doesn't mean you couldn't have a Frankish succession turning, not exactly to "primogeniture" (that AGAIN, isn't an ideal form of succession existing from all eternity, but the result of a long political evolution) but maintaining as long as possible a monarchic figure (as in, only one king).
Interestingly, you'd need a frankish-screw for that.
See, the historical model lasted because it worked : give a king monies and a border to defend, and each one made the Frankish conquests and sphere of influence growing up to swallowing up Western Europe from Atlantic to Bohemia.
You need to break Frankish rise, in order to make it. Making Clovis an Homean may help things work out : Gallo-Roman clergy wouldn't support him nearly as half they did IOTL, and you may end up with the maintain of a Gothic overlordship more easily in southern Gaul.
Weakening Frankish power, and possibly going to similar inner crisis that knew Visigoths IOTL may as well end up into a formally elective kingship,
counciliar or not, that is far more easily monarchic (also clearly more unstable).
Less territory (meaning less division of it), more important foreign pressure, inner crisis, may end with a monarchic elective frankish kingship, possibly evolving afterwards with a "natural" succession of one kingship (basically, what happened in Early Capetian France).