I kind of like "Cadeau" (gift, present) as another option.I love Cosette. Cosy for short.
Or take some meaningful words and try translating them:
-The day of the week you first saw her (or the day you pick her up)
-The French version of her dam's name
-Her colour (Noire would be a bit odd, but I love Blanche)
So exciting!! Congratulations ?