Is it possible that your primary title upon creation of the Empire of Francia was elective? When you create a higher title, primogeniture and seniority carry over, but elective does not - it always reverts to gavelkind. The other option is that the Empire of Francia did indeed come with preset gavelkind.
The question really is if the empire law (gavelkind) overwrites the kingdom laws (elective and primogeniture).
If it doesn't, this means that your possessions (counties, duchies, kingdoms) which fall under primogeniture will go to your eldest child (depending on agnatic, agnatic-cognatic, cognatic). Your possessions (counties, duchies, kingdoms) which fall under elective will go to your chosen successor (in case you chose your eldest child, the same as under primogeniture). Your possessions which fall under gavelkind (the empire title) will be split among your children. Since this is only a single title, though, this will indeed be transferred to your eldest child.
If the empire law (gavelkind) does overwrite the kingdom laws (elective and primogeniture), this means that all your possessions (counties, duchies, kingdoms and the Empire title of Francia) will be split among your children (depending on agnatic, agnatic-cognatic, cognatic). This means that your counties, duchies and kingdoms will be split according to gavelkind, and the Empire title of Francia will go to the eldest child.
It's slightly more complicated than that, because while your possessions will be split according to gavelkind, the counties and duchies which are not held by you personally (i.e.: you don't hold the title yourself) but answer to a certain kingdom will go with that kingdom. This is also the case with primogeniture and elective, but seeing as your possessions aren't being split up (they are all held in personal union by the same person) this isn't an issue. If the gavelkind law (empire) overwrites the primogeniture and elective laws (kingdoms) then this means your kingdoms will be split up between your children and the duchies which are not held by you personally will go with the kingdom they're attached to.
Personally, I'm not 100% sure if the empire law overwrites the kingdom laws. It should be easy to tell, though - the game warns you if your inheritance will be split up (per gavelkind succesion). Look for this button in the top centre of the screen, and go over the portraits of your children in the family screen. If this button does not appear, and if none of your children barring your eldest child (primogeniture) and chosen successor (elective) have the title "heir of [...]", you should be safe. But, again, I'm not 100% sure - I've never been in this scenario myself.