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First, I love [name_u]James[/name_u] and [name_m]Oliver[/name_m]! I love [name_f]Josephine[/name_f], but I agree, not with [name_u]James[/name_u]. But I have some other classic girls names you might consider 
[name_u]James[/name_u], [name_m]Oliver[/name_m], and…
[name_f]Amelia[/name_f] or [name_f]Emilia[/name_f]
[name_f]Charlotte[/name_f]
[name_f]Cora[/name_f]
[name_f]Emma[/name_f] (Rarely do I suggest my own name, but I really like it with [name_u]James[/name_u] & [name_m]Oliver[/name_m].)
[name_f]Katherine[/name_f]
[name_f]Lucy[/name_f]
[name_f]Margot[/name_f]
[name_f]Matilda[/name_f]
[name_f]Rose[/name_f]
[name_f]Sophia[/name_f]/[name_f]Sophie[/name_f]
I think [name_f]Josephine[/name_f] nn [name_f]Posie[/name_f] would be fine. S/O and his brothers’ (and one sister’s) names all start with J, but then two sisters have names beginning with other letters. I don’t think it’s that clashing. Especially since [name_u]James[/name_u] and [name_f]Josephine[/name_f] don’t really share any sounds.
For suggestions though;
[name_f]Eleanor[/name_f] “[name_f]Elsie[/name_f], [name_f]Nellie[/name_f], [name_f]Ellie[/name_f], [name_f]Nora[/name_f]”
[name_f]Elizabeth[/name_f] “[name_f]Elsie[/name_f], [name_f]Lillie[/name_f], [name_f]Betsy[/name_f], [name_f]Libby[/name_f], [name_f]Bessie[/name_f]”
[name_f]Penelope[/name_f] “[name_u]Penny[/name_u], [name_f]Nellie[/name_f]”
[name_f]Beatrice[/name_f] “[name_f]Bea[/name_f]”
[name_f]Martha[/name_f] “[name_f]Maisie[/name_f]”
[name_f]Matilda[/name_f] “[name_f]Millie[/name_f], [name_f]Tilly[/name_f]”
[name_f]Lucille[/name_f] “[name_f]Lucy[/name_f]”
[name_f]Charlotte[/name_f] “[name_f]Lottie[/name_f]”
[name_f]Emmeline[/name_f] “[name_f]Emmie[/name_f], [name_f]Millie[/name_f]”
[name_f]Olivia[/name_f] “[name_f]Liv[/name_f], [name_f]Livvie[/name_f]”
[name_f]Isabelle[/name_f] “[name_f]Ellie[/name_f], [name_f]Belle[/name_f]”
[name_f]Isadora[/name_f] “[name_f]Sadie[/name_f]”
[name_f]Alexandria[/name_f] “[name_f]Annie[/name_f], [name_u]Lexie[/name_u], [name_u]Allie[/name_u]”
[name_f]Georgia[/name_f] “[name_u]Georgie[/name_u], [name_f]Gia[/name_f]”
[name_f]Abigail[/name_f] “[name_u]Abbie[/name_u]”
[name_f]Arianna[/name_f] “[name_f]Annie[/name_f]”