However the duchy of Burgundy under the Capetian house of Burgundy wasn't, in fact they could pass inheritance through the female in the absence of a male heir; this situation was basically claimed by Valois-Burgundy (for obvious reasons).
Actually, the Capetian House of Burgundy was retro-actively considered as an apanage; while neither the concept or name was used in the early XIth century.
It's partially due to the cession of the Duchy to Henri by Robert II, and when this one became his successor, to his other son Robert.
You already have the precedent of a duchy that was gaven to a son that wouldn't be transmitted the crown (especially in a time where sons were crowned during their father's reign).
(It's interesting to note that it was transmitted as such even before Hugues Capet came to power)
Generally, it's understood as a proto-apanage, even if it wasn't obviously considered so : Valois had good counter-arguments on the first Capetian Burgundy nature.
Also Burgundy was much more than the duchy of Burgundy proper, many of those titles were gained by inheritance or even purchased.
Indeed, but I'm not sure I said otherwise in the last part of my post.
This basically means that the heiress of Burgundy will lose Burgundy proper, but can hope to retain the French fiefs Artois, Flanders and Charolais and their Imperial Possessions (the duchies of Lothier, Brabant, Limburg, and Luxemburg; the county palatine of Burgundy; the counties of Hainaut, Holland and Zeeland; the margraviate of Namur).
I doubt it could hold onto that as easily you picture it : Valois France was at this point strong enough to enforce a large claim on Burgundy (especially when it went to regions not only under french suzerainty, but whom acquisition was made by the duke on Valois). Artois especially would be too important strategically for not being largely disputed, even if holding on Flanders proper seems far less certain to me (you'll need Valois to make far better offers than any rival, and that's not something they could pull easily).
As for county palatine, I'd think it would essentially depend from the imperial good will : if they're interested on preventing a too a powerful principality in the western part of the empire, more than breaking Valois (which, admittedly is open to discussion), they could support partial claims from France on both Burgundies.
Or Valois trying to trade it in exchange to renouncing Flanders, but giving they couldn't have really hold on it in first place...Depends if Lorraine/Burgundy is willing to go for attrition war or not.
Admittedly they could be more interested into splitting Artois in half, with N-W (Saint Omer, maybe Thérouanne) for Burgundy-Lorraine and S-E (Arras) for Valois
We discussed it some time ago, and I think I'm generally sticking with what I tought would be the situation with such marriage with Lorraine, basically
this (As said above, I'm dubious about the Free County. Not that Valois would be particularly interested on it, but they'd be annoyed at any kind of geographical encirclement)
Especially since the king of France at this point is also facing opposition from most of his high nobility.
Giving most of this opposition was fueled by Charles we could certainly not seeing it appear. Without him, there's only Brittany with enough strength and political power to take its role, and they had neither the ressources or the possibilities Burgundy had, and the League wouldn't benefit from the encirclement Charles made possible : Bretons would be the main priority and probably crushed hard.
In any case 'Burgundy'** will shift from an French vassal holding Imperial fiefs to an Imperial vassal holding some French fiefs.
I don't know if they would get unconditional support from HRE : Imperials would prefer to clean off the mess, appear as arbitles of peace and prevent a too powerful entity in their western periphery. Preventing Valois to turn all Borg is one thing, but they'll keep in mind their own interest.