So, since I always been intrigued by the initial concept of a french hre, while surely not knowledgeable enough for such huge divergence I will give you my two cents, thought on those last days, about it:
- Robert, king of West Francia wins the battle of Soissons and lives for some more years.
- He lives his remaining time consolidating power (lotharingians, normans, occitans, ...).
- Hugh, the Great/the White, his son doesn't get some thoughts of unholiness of usurpation nor he needs to fly low in front of his 'peers' in the realm.
- We need to find him another wife that would give him scions. Here is where I've got some difficulties: there aren't many good matches as far as I know. But, his father being alive, we could have him trying to consolidate his base, marrying him to the sister of Rudolph, duke of Burgundy, son-in-law of Robert, brother-in-law of Hugues.
(- Down the line, Rudolph dying childless, we would get some interesting conflict between the two Hugh, the White and the Black.)
- Burgundian (Kingdom) opportunities would be grasped by Hugues (now King of West Francia): first by meddling with arbitration between Rudolph II and Hugh of Arles with control of Viennois under Charles-Constantine and later taking care of infant Conrad III at court formally controlling his kingdom and as adult making him marry one of his daughters.
- There would be, before that, some lotharingian revolt on the death of Robert and coronation of Hugues with Henry the Fowler's intervention, but those would be put down bringing with time (and other tries) the end of East Francia's ambitions on the west bank of the Rhine (pre-modern era).
- Italian adventures would be something for Hugues' Son, let's call him Robert. He would marry Adelaide, stopping his first descent on the peninsula early due to the death of his father and returning home to be crowned. Having stabilized his realm he would descend a second time and gain the imperial crown.
- And so we got it.
East Francia on the other hand should be screwed a little so to have a unified modern state by the XV century and not before. Let's say Henry the Fowler's defeat in Lotharingia and a premature death of his son Otto would get a full of concessions second son Henry on the throne and a Voglers dynasty slowly reaffirming royal power.
Now, for social/political differences, that ain't easy.
Seems to me there is a Robertian preference to devolve temporal power to secular princes rather than ecclesiastics. I think on the long term it would mean way less ecclesiastic demesne on imperial immediacy than OTL. How does this interacts with the Cluniac movement ? What about Gregorian reforms ? Would there be an investiture controversy on which terms ?
Let's say we mimic OTL to not enter a vast uncharted land as blind men.
- Robertians die out.
- A second imperial family comes to power whose reign is characterized with papal conflict.
- They die out.
- A third family ascends to the imperial crown. Their power base is an oversized duchy of Burgundy (ducal burgundy, county, cisjurania, transjurania, valentinois, foulcalquier) and start a struggle to retake control of Northern Italy.
- Italian polities are divided: on one side the "Cusci" (from the castle of Couches, place of birth of this imperial family) siding with the emperor, and on the other the "Curtini" (from the Courtenay, imperial rivals) siding against imperial ambitions and with the pope. Such sides would exists through the XII, XIII and XIV centuries independently from imperial meddling.
- Inheritance of the Norman Kingdom of Sicily (would it still exist ?) brings other changes in italian dynamics. Imperial Sicilian King could bring a following of occitan troubadours (or oil trouvers) influencing way more the beginning of italian cultural vernacular development successively defined by the tuscan school of stilnovists.
I do really think there are ways for having an italian language developing as a member of the francophone family. As there is a real possibility to get a more Oc french rather than Oil. But such an idea necessitates way more thoughts for it to be elaborated satisfyingly.
As for other internal political developments: the end of this third line gets us an Interregnum and a disintegration of their duchy with the rise of a multitude of tiny polities rich enough for autonomy on the commercial road from Italy to Flanders/Lowlands.
You could have:
- a Switzerland-Like confederation on the Alpine passes.
- A burgundian league.
- Some obscure count of Albon ending the Interregnum taking back from the Kingdom of Navarra "illegally" held lands in the Toulousain conquered during an unsanctioned crusade on catharism and giving them to his family later becoming a power player in modern era Europe.
- A golden bull with Navarra, Anjou, Normandy, Ecclesiastic Sees of Reims, Cologne and Trier, and some prestigious count of Lotharingia as Electors.
Still, there are way more entities and movements to relate to such a divergence:
- Norman conquests.
- British involvement.
- Iberian peninsula.
- Hungary, Poland and Bohemia (possibly ending as German Brittany) for East Francia.
- Crusades.
- Templars (I would like to put them on a conquest of Valencia).