Bryony/
BrionyCambrie/
CambriaBriseis- Her experiences were horrible in the Iliad though.
BreanaBrighidBrigidaAdelbrie
Brisa
Briseida
BreindelBriallen- pronounced bree-AHSH-en in Welsh
Brita
You could always use
Bree as the full name since it is the name you prefer.
Brigh is an Irish name meaning "power," and it has the same pronunciation.