I feel like the problem about the bats, and why I don't think they'll ever truly get along with Jason, that they won't compromise themselves for Jason.
Jason is the one that has to change if he wants to be in the family.
All the bats can talk about how they want Jason to come back home, but with conditions on Jason's end. They just won't acc pt him otherwise.
They don't bother to do anything to mend their relationship either or at least genuinely, seeing things from Jason's end. Why Jason does the things he does? They don't bother to think why.
Like it's so obvious (to readers at least), that Jason has become disillusioned by heroes and the justice system, he no longer believes that they're useful/efficient, that all it's doing is perpetuating crime, because criminals are getting if scot-free, not facing the consequences of their actions that have massively impacted innocents lives. And that's why, he kills/killed people, because sure it's not right, but it works (although realistic it's more short-term), whereas the bats methods doesn't as much.
And despite this, the bats do nothing. They parade around saying they want to stop crime, but violence only breeds more violence (and yes this does also apply to Jason). Even if they are helping Gotham ad citizens (e.g. health care, giving jobs, donations to charity, whatever), it doesn't seem effective. Plus Bruce puts uses more money for his vigilante scheme than actually helping people (imo).
Like crime occurs in places with poverty, that's basic crime 101 (increase in poverty/inequality = increase in crime rates), and yet I don't see Bruce trying to do anything to help Crime Alley change back into Park Row. What I do know, is that Batman only visits the place once a year and is just throwing himself a pity party (he visits on the day his parents died - ref to when Jason first appears post crisis).
Like they just can't understand Jason, can't see where he is coming from (I would say that Duke and Steph are the one who's most likely understand). But the things is, is that they don't even try to understand where Jason is coming from, just condemning him for killing people.
And it's not even that, like they themselves don't have to kill, but I feel like if they just understood and accepted Jason's methods are different, not condemning him, then maybe imo, they could be family. Because all they're doing is pushing their morals and methods onto a man who clearly doesn't agree, but does so anyways just to fit in with a family.