I never let a "common nickname" ruin a full name for me. I personally love a lot of full names and not the expected nickname. In fact, there are plenty where I wouldn't use a nickname when everyone would have the tendency to shorten it.
I've met plenty of people who would expect to shorten their names and tell people they wish to be called by their full name. (Names such as
Michael,
Nicholas,
Patrick,
Joseph,
Katherine,
David, and many more.)
I say use
Josephine if you want. Just tell people you are calling her
Josephine or
Joey (or whatever nn you decide on). Family will know, and friends. But remember, she may like
Josie when she grows up and may want to be called
Josie instead.