After Jason’s death we know Bruce was going through an extremely dark time. So dark that little Tim saw him and went, “you need someone to watch you. You’re my child now”
And from there he kinda chilled out, he had a Robin again to keep him grounded
But I imagine after Jason resurrection he became more set on his no killing rule. Because he didn’t even know the Red Hood was his son at the beginning. Even as he committed these bloody crimes Bruce still loved Jason, right?
So even as he has criminals that are doing awful things he just can’t see them as mere criminals, he sees them as brothers/sisters, sons/daughters, wives/husbands, nieces/nephews, someone. Someone who if they were to die would leave a family behind
Now imagine a world where Bruce did snap after Jason’s death, he started killing. Then this Red Hood comes to town and he kills him too. Now either imagine Bruce never takes off that helmet and continues on never knowing his killed the child he lost, or a world where he does take the helmet off and is left with what he has done
Or in another where the Red Hood comes to town and the Batman finally snaps and kills for the first time, only to realize it was his son