You could also name her Josee which is kind of middle ground. But I agree. Of you both like josephine, put that on her birth certificate and her daily name on her stocking
Put Joey on the stocking and leave it Josephine on the birth certificate. That way when she grows up she can be a go-go dancer (a la Joey Heatherton) or President of the United States (in which case one would assume she might go by Josephine). If you were planning on calling her Joey anyway, he should not be too distressed about what is on her birth certificate. Josephine is a lovely name -- lest we forget, French historical as well, for one of the most celebrated women in history.
I say leave it Josephine Clara. She can go by Joey so not too long but I don't think it is anyways. My cousin named her daughter Josephine Catherine. That's way longer.
That's completely up to you guys.
My bf has all short names. I don't really like that. I tend to give short first names with long middle names or vs vsa. Its all up to you preference. . Waiting until baby is born and tooling at her little face will help you decide if she looks more like a Josephine or a Clara.