It seems like framing it in all out war is maybe a little drastic. To have Batman be able to go along with a group like that seems kinda far out and not really good for the character long term. I haven't read Kingdom Come tho, so I'm not referring to that, but just the scenario you're putting forth.
It sounds like a What If? or an Elseworlds but just not good for ongoing characterization.
To me, it seems like it'd be enough to have people freaked out about the appearance of Superman and lots of people really casting him in a suspicious and untrustworthy light. From the government, the media and Luthor too, would make sense.
But it would just be this overall feeling of distrust and wariness. And Batman wouldn't be part of this, but he'd just be doing the sensible thing and suspicious himself but not all witch hunt about it. and independently prepare to maybe have to fight him if it comes to that and seek out a meeting with him.
at at this meeting, neither of them is too big on the other one in this context and things just escalate naturally to a fight. like maybe there's some other incident that makes Batman even more uneasy with these super powered people running around.
maybe we're talking about the same thing, but i think making it a 'war' is pushing it to lines that don't need to be drawn, really. or that the story would be better off for not drawing. and also it would call into question, i think, to have Batman ever allied with Luthor. he doesn't realize what's up with this guy?
Superman would have to actually be a straight up villain for Batman to go that way, I think. Batman should have a good bs detector, just in his own instincts.