I love Posy as a nickname for Josephine, in fact I'd be inclined to use it all the time, i.e. the child would use it at school, sign it as her name, and Josephine really only be used on the birth certificate.
Is it worth having Josephine as a name to fall back on? I'm thinking that would be the case, as, for instance, I love the inspirational figure Josephine Baker. Or does Posy stand alone. Would you put just Posy on the birth certificate?
Oh, and what would go well as a middle name?