Batgirl is Barbara's legacy, and Barbara's the one who gets any say in who gets the costume. Steph convinced Babs she was up for the job. If Bruce had been alive and had a problem with it, Babs would have made her one of her agents and told Bruce where to stick it, the way she did with Helena.
And yes, I am also peeved as hell about how everybody seems to have forgotten Cassandra ever existed.