I would choose: Isabel (I also like Isobel), Violet or Phoebe! Kate’s a bit too simple/stark with your other, pretty names but it would be my fourth pick. Kate does have a lot of punch!
Emilia/Amelia is too close to Emma as others have said, especially as the next baby in birth order. Ruby and Lily are a little close to Lucy - and I also see them very much “the youngest child” names. That’s not a dealbreaker, as they can be used for any child - just that often the youngest child gets a more “cutesy” name.
Other names:
Mary
Rose
Iris
Ivy
Tess
Jane
Fleur
Freya
Helen or Helena (nn Leni)
Margot
Sylvie
Astrid
Ingrid (nn Indi)
Annika
Frances
Beatrice
Ophelia (nn Lily) - but note the Taylor Swift effect (likely to be very popular in 2026) - edit: also Ophelia is probably too close to Sofie
Some middles:
Seven
Septima
Eden
Ever
Alma
Livia / Liv
Finn (play on fin meaning final)
Seraphina
Good luck!