Bernice is okay, it just sounds a bit old. I like
Bianca better.
Bernice would be alright but only use it if you really love it and don't mind a little critisisim from others.
As for nicknames there's the obvious
Bernie, then what about Nice (like nee-s or nee-see). Ice or Icy would also work.
Berry would too.
I hope this helps!
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Ciaran Douglas, Aidan Stone, Simon Andrew & Amy Frances, Julia Mary, Elena Beatrice