There have been so many incarnations of the Bat Family. It’s sorta become like mutants/X-Men in Marvel - too many members to count and not all fit timelines so neatly to put it nicely. Quite a few characters have even taken on more than one Bat Family mantle. If it were up to you who would be in the Bat Family and who would be who? Anyone not used either does not exist or not part of the team.
For me:
Batman (Bruce Wayne) - obviously a must. He started the whole damn thing. The only thing I would say is sooner or later Batman must die and rather having someone take the mantle I would rather see the fam take over in his stead.
Alfred Pennyworth - it’s ■■■■■■■ Alfred okay? There is no family without good ol’ Alfie.
Nightwing (Dick Grayson) - the first and former Robin (Boy Wonder) and my absolute favorite character. A must. After quitting as Batman’s sidekick and forming the Titans while in college he becomes Nightwing when learning Batman had replaced him. After graduating college, the Titans would disband for a short time to build their own lives and Dick would take over for Bruce upon his disappearance for a short time, training a new Robin and Batgirl with the help of friend…
Oracle (Barbara Gordon) - the first and former Batgirl would become Oracle after getting paralyzed by the Joker. An event that would cause then boyfriend, Dick to quit as Batman’s sidekick. Also the leader of an female hero team Birds of Prey.
Tim Drake - The third Robin and Dick’s other half to the Dynamic Duo. I chose Tim for this spot instead of Damian for timeline purposes. Plus having Dick bring in a new Robin, especially Tim who finds out their secret identities in Batman’s absence and having to convince him that Tim is worthy upon his return is just a better use of his story. And then having Tim ousted as Robin by the next person for not being choses by batman himself just makes it all the better. He would need a new name besides Red Robin though otherwise he will be a perpetual Robin who is not Robbin and it just doesn’t work.
Robin (Damian Wayne) - is there really anything to say? Batman finds out he has a bratty, entitled little prick who forcefully takes the mantle of Robin from Tim.
Batwoman (Kate Kane) - appearing in Gotham shortly before Bruce’s disappearance and brought in by Dick when he takes over as Batman for a little bit.
Batgirl (Stephanie Brown) - Barbara encounters Stephanie acting as Spoiler while in a hacking war against Cluemaster and makes her the next Batgirl.
Yes, Red Hood exists (no matter how much I hate the character). Really wish I coulda axes his character but his death alone is one of the most important stories in Batman mythos. He’s just not part of the team because he goes against everything Batman stands for. Huntress also exists but she’s Birds of Prey not Batfam.
Edit:
Thinking about my little timeline and version of the Batfam I’m gonna add Luke Fox as the new Batman after Bruce’s death. I think that’s a better use for his character.
Imma axe Cassandra Cain, Duke or “The Signal”, and “Blue Bird” 
from existence. There’s already a child raised as an assassin who turns to the heroes with Damian. We don’t need another. As for Duke, while I am curious as to the goings on in Gotham during the day, Duke is absolutely the wrong character for that. Plus, his powers make no sense, his costume is ugly, and we don’t need another “former Robin” which I think is incorrect as he was never truly Robin. And I don’t like the Nightwing rip-off that is Blue Bird and just don’t think she fits.
Catwoman, I don’t see her as part of the Batfam considering she’s a thief. Like Batman’s whole thing is “No stealing. No Killing.” so I don’t really think she has a place so same boat as Todd.
Azreal, I don’t understand enough to even consider him or think about him existing so axed I guess…?
I think I’ve gone through just about everybody now.
