I've always pronounced it beh-reh-NEES, like Bérénice Bejo. Another Bérénice, Bérénice Marlohe, is going to be the next Bond girl (her character is named Sévérine...swoooon!).
Ber-ə-NIE-see is new to me too. It's not an intuitive English pronunciation, and I don't know that you'd run into it all that often, but I also don't think people would assume bər-NEES as regularly as you'd like either. That middle E seems to beg to be pronounced differently than Bernice.