It seems very likely that LaBarre is not in the old region of Picardy. The name Picard means a native of Picardy and was probably given to an ancestor who moved from Picardy to someplace else. So it would not make much sense for a family in Picardy to use the surname Picard to distinguish itself from other families in their village in Picardy.
In the last scene in "Family" Rene Picard is outside at night when a meteor streaks across the sky. The constellation Orion can be see above the horizon. Thus it seems probably that "Family" happens in winter, that Orion appears over the southern horizon, and that La Barre is far enough south for Orion to be that far above the horizon. It is possible that La Barre is in Corsica or in land reclaimed from the Mediterranean sea floor in future times.
Is La Barre in Haut-Saone at 47°24′20″N 6°10′50″E far enough south for Orion to be that high in the sky? https://en.wikipedia.org/wiki/La_Barre,_Haute-Sa%C3%B4ne1
If not, are there any other present day La Barres far enough south, or must we assume that Picard's La Barre was founded sometime in the future?