In my experience, [name_m]French[/name_m] names during the early 1800s were very similar to Puritan names in the US - either biblical classic choices, or off the wall, unique word names. Going by that, here are my suggestions.
Father: Works for the government, dislikes the revolutionary groups, gentle with this wife, but ignores their children.
[name_m]Alexandre[/name_m], [name_m]Gilles[/name_m], [name_m]Gaspard[/name_m], [name_u]Maxime[/name_u], [name_m]Cesar[/name_m], [name_m]Xavier[/name_m]
Mother: Kind, gentle, very sickly and deeply devoted to her husband.
[name_f]Marie[/name_f]-[name_f]Jeanne[/name_f], [name_f]Marie[/name_f]-[name_f]Madeleine[/name_f], [name_f]Charlotte[/name_f], [name_f]Jacqueline[/name_f]
[name_m]Son[/name_m]: (19) A woman’s man, ignores his younger siblings, was his father’s pride and joy.
[name_m]Louis[/name_m], [name_m]Etienne[/name_m], [name_m]Jacques[/name_m], [name_m]Henri[/name_m] [name_m]Aurelien[/name_m], [name_m]Gabriel[/name_m]
Daughter (the protagonist): (17) Kind, good with children, witty, very independent.
[name_f]Marie[/name_f], [name_f]Amelie[/name_f], [name_f]Helene[/name_f], [name_f]Isabelle[/name_f], [name_f]Genevieve[/name_f], [name_f]Gabrielle[/name_f]
Daughter: (14) Sweet, sickly, dependent.
[name_f]Agnes[/name_f], [name_f]Adrienne[/name_f], [name_f]Blanche[/name_f], [name_f]Colette[/name_f], [name_f]Aimee[/name_f], [name_f]Claire[/name_f]
[name_m]Son[/name_m]: (7) Naughty, happy-go-lucky, demanding.
[name_m]Gaston[/name_m], [name_m]Bastien[/name_m], [name_m]Charles[/name_m], [name_m]Edouard[/name_m], [name_m]Christophe[/name_m], [name_u]Jean[/name_u]
Daughter: (4) Caring and friendly.
[name_f]Christine[/name_f], [name_f]Delphine[/name_f], [name_f]Elisabeth[/name_f], [name_f]Victoire[/name_f], [name_f]Henriette[/name_f], [name_f]Angelique[/name_f],
A key thing to note though, many names were hyphenated (and still are) in [name_m]French[/name_m] cultures - most commonly [name_f]Marie[/name_f] for women, and [name_u]Jean[/name_u], [name_m]Charles[/name_m] and [name_m]Louis[/name_m] for men. With women, at least, sometimes women would all bear the “[name_f]Marie[/name_f]-”, and formally go by the second half of the hyphenated choice.
Example;
Mother: [name_f]Marie[/name_f]-[name_f]Angelique[/name_f] “[name_f]Marie[/name_f]” (usually the eldest used [name_f]Marie[/name_f], from my understanding)
DD1: [name_f]Marie[/name_f]-[name_f]Claire[/name_f] “[name_f]Claire[/name_f]”
DD2: [name_f]Marie[/name_f]-[name_f]Amelie[/name_f] “[name_f]Amelie[/name_f]”
DD3: [name_f]Marie[/name_f]-[name_f]Jacqueline[/name_f] “[name_f]Jacqueline[/name_f]”
DD4: [name_f]Marie[/name_f]-[name_f]Christine[/name_f] “[name_f]Christine[/name_f]”