Metamorpho is a Bat-Family member. The Outsiders weren’t like the JLA, where Batman was just a member. Batman was the creator and leader of the Outsiders. They were stationed in his penthouse. You don’t get more Bat-Family than living in Batman’s home.
I prefer Oracle to Barbara as Batgirl.
Oracle
Casandra Cain
Vesper
Catwoman
Batwoman