I feel like Josie/Josephine might be something to be cautious of with your H's similar first name- not only the close name/nn but the fact that you might be giving her the impression that you really wanted a boy to be Joseph Jr. and you settled on a feminine version of it instead because you ended up with her. I would definitely go Josephine instead of Josie if you are set on that name though, it ages better and gives her a nice full name to use professionally, even if you only ever call her Josie.
Both Josephine and Lila are nice, but I'd go with Lila in this case.