Louis' grandsons share the empire after a civil war. Helped by sons-in-law of Louis. And rogue noble more or less tied to Carolingians.
The collapse of Carolingian empire isn't due to a familial quarell only : the empire was unified OTL after 843 but it didn't lasted as you had much more tendencies present.
Carolingian empire stability was based on a gift-based economy. Conquest and trade managed to maintain the loyalty of the Frankish (I use Frankish here for commodity, but it include Bavarian, Aquitain, Saxon, etc) nobility, that was formed along a vassalic clientelist relation.
When conquest stopped and trade declined, Frankish nobles hadn't many choices : focus on their lands and try to grab what the neighbors had.
Bad harvests due to climatic changes probably helped as well.