I can honestly say that I was in a similar situation with one daughter. Her name starts with Bri and before we named her I said we would never call her Bri/Bree, now she pretty much goes by Bri and I can honestly say that now I love it and that it goes with her fun and breezy vibe. It definitely grew on me. That’s not to say that Josephine will definitely be Josie someday, but perception does change with the personification of your adored daughter (as Abby already mentioned).
Also, we’ve had a Josie (named just Josie) in my oldest class who is 15 all through school, and not once did anyone mention or associate her with Josie from the movie. And I’m 35 so I know exactly the movie you’re talking about, but again no “Josie grossie” ever crossed my mind. I hope that makes you feel better about that situation or the possibility of it.
Congrats and hope this helps!! :)