Being that [name_u]Frankie[/name_u] is already a nickname-nsme I don’t think people would turn to [name_u]Fran[/name_u] or [name_f]Franny[/name_f].
I like [name_f]Florence[/name_f] from your list a lot. I think it strikes a balance between both of your tastes, and dad could probably still get away with calling her [name_u]Frankie[/name_u].