I like
Breda. And
Brighid - although it sounds almost like
Bree, so it would probably not need a nickname... Or maybe some variants of
Brighid will do -
Bridget,
Brigid, or
Brigida, the Italian and Spanish equivalent.
Oooh, I totally forgot - would about Greek
Briseis? That's uncommon and
Bree would be a lovely nickname!