I always thought that
Berenice was pronounced bər-NEES like
Bernice or beh-reh-NEES. It wasn't until I read about
Berenice on this site that I encountered ber-ə-NIE-see. ber-ə-NIE-see sounds pretty to me, but I think that there would be fewer issues with bər-NEES or beh-reh-NEES.