Bruce Wayne

Jason Todd - Adoptive son

Tim Drake - Adoptive Son

Damian Wayne - Son

Cassandra Cain - Adoptive Daughter

Bruce Wayne's backstory is the same, duh, his parents die in crime alley, bruce realizes he was a little shit and goes all broody. As a kid Bruce develops depression and maybe a bit of an anxiety disorder. Regardless, Alfred looks after him and makes sure he has a good childhood. Bruce makes little to no friends throughout his childhood, instead focusing on studying and filling his brain with as much information as possible. He gets his degree in psychology straight after high school, becomes study buddies with Harleen Quinnzel, and then right after college he fucks off to Asia to master martial arts and get himself into the peak of physical perfection.
id say hes...25? 26? when he starts doing vigilante nonsense. he doesnt announce himself as Batman yet. he's also not hiding it from Alfred, they just kind have a mutual understanding that he dresses like a bat and then beats up nere-do-wells in the middle of the night
alfred only really gets in on it like half a year in where hes like
"you know...you could use an assisstant..."
"yeah, having someone in the mansion to make sure i know where im going is a good idea"
"maybe someone with war experience...and a supercomputer..."
"do you want to do that, alfred"
"very much so, master wayne" (big, 'i crave violence, master wayne')
i think alfred invents the batcave. bruce wayne is an uncreative hardass but his dad? his DAD is a whimsical bastard. alfred turns the caves under the mansion into his base and goes "GET IT! CAUSE YOUR A BAT!"
and bruce is very dissappointed but very proud
(thats also why all the things are named bat-grapple, batarang, batomobile, bat-wing, bat-computer. bruce is just super uncreative)
maybe his rogues gallery starts building around here too. idk who he'd start off with, probably organized crime bosses? Penguin, Black Mask, you know the drill
the first metahuman threat is probably Ivy or Croc or smth
though when bruce comes pu with the "batman" moniker is DEFINETELY after a personal encounter. Harvey Dent, an old college friend, getting half of his face burned off after bruce failed to save him from an assassination attempt? thats gotta fuck bruce up bad. hes gonna dive headfirst into this
though he does make allies along with his MYRIAD enemies. his shaky alliance with Catwoman built entirely on mlm/wlw hostility, the several sidekicks-turned-adopted children, etc.
also, when it comes to his rogues gallery, while batman uses force, he DOESNT think his enemies are just irredeemable villains
as a mentally ill adult man, bruce 100% recognizes that a majority of his villains are just people that have been horrifically wronged by the system (Ivy's research and work being destroyed in favor of an oil plant that ruined her dreams, Croc being used as a human experiment, Harley having been physically and mentally abused most of her life, scarecrow having his work discounted along with his years of mental abuse and neglect, clayface being abandoned and neglected by hollywood after getting "too old", etc.) and wants to see them recover
batman might be cold and harsh, using force to stop bad guys, but he's still empathetic. hes still a human person
i think if i were to write a batman id want to place him somewhere between animated series batman and adam west batman.