The POD here is that Lothair dies sometime before 843, leading to an alt-Treaty of Verdun in which the Frankish Empire is divided into two realms, with the River Rhine being chosen as the border between them. How does this affect subsequent European history?
If Lothaire dies in 843, then you'd end up with a really complex situation : aside Charles the Bald and Louis the German that without a common foe would probably fight each other as they did eventually, you'd still have to deal with Pépin of Aquitaine (who still had control of a fair part of Aquitaine), Louis of Italy in...well Italy, and the two others sons of Lothar.
The Frankish aristocracy
already prooved that it wasn't going to suicide itself for the claims of the Carolingians, which is partly why Charles and Louis reached an agreement, but you'd end up with a chaotic situation where alliances are likely to be reversed, and complexified at least in a first time, with the order issued from
Fontenoy being at best contested.
It's unlikely to me that you'd end up with a dual partition of the Empire : at the contrary, I think you'd end up at least with something more complex, an assortment of various alliegences and loyalties, that would require more fights (with a wary and weary nobility) to assert. Maybe something along the
Ordino Imperii settled in 855, depending on how (shifting) alliances unfolds.