Picard is from 'France' as is stated in the show....however that does not mean 'France' in the modern sense. It is simply a region, a geographical area. NOT an independent, autonomous body.
Problem with that is, once France ceased being a political unit/country, how much longer would little Le Barre continue to be "
We seen in the past few decades where a country has ceased to exist, broken up, that the people fairly quickly stop identifying with the former large unit.
Most of those entities--Yugoslavia, Soviet Union, Czechoslovakia--were created late in the process of modernization when various allegiances to smaller nationalities had already been made. On the other hand, four decades of separation between eastern and western Germans created cultural divergences, but it did not undo the sense that they belonged to what should be a single entity. Would La Barre remain French? Franche-Comté is, in many respects, very Swiss, so I doubt that the disappearance of France as a political entity would make move any further in that direction. If their sense of Frenchness is drawn from their local circumstances and culture, then they will likely continue to identify their local culture as being French.