Big problem is that the imperial title is pretty mighty juju. It won't just go away. You'd have to have the sons screw up majorly to have it drop out of use.
If it happens, the most likely fracturing lines are going to be the old dukedoms and kingdoms. I can't see any court noble taking control over some territory - they lacked support and access to resources without the imperial power to call on them. Most likely the entire court structure will be discredited - this could extend to the church, where Eigenkirchen and traditional bishoprics could see a new boost of influence against the reform monasteries and royal chapel.
You could well see Saxony breaking off entirely from the Carolingian tradition and hitching its fortunes to Denmark. Thuringia, Bavaria, Gascony, Bretagne and Aquitaine all have their own local traditions of government and could well simply stop deferring to the royal centre. Francia is likely to stay in one piece. Frisia may stick with Francia, but there's still a chance it will go its own way. The Lombard kingdom, of course, can always revert to being its own kingdom, as it did later. If it happens early enough, that could be bad news for the papacy - the 9th century popes weren't exactly models of probity or political genius.
Personally., I think the imperial title will stay in play even if the Carolingian family loses its prestige.