Yeah, as was already said, I honestly don't think DC really cares.
When was the last The Question ongoing? Was it the Dennis O'Neil and Denys Cowan series from the 80s? There's been a few limited series since then, but any ongoings?
If not, then there's your answer there.
You're right in what you're saying, but thinking about it, I don't believe Jace was really that active in Gotham City for that long.
Five issues in real time, but in comic time it may have only been a few days.
I think the "Fear State" tie-in was issue #3, and he left for New York to go with his mother and Tiffany, because his sister Tamara was going to New York for physical rehab, and that happened at the end of issue #5, so that was the last time he was in Gotham. And he only went out as Batman for the first time a few days (roughly) before the "Fear State" tie-in.
The police saw him and everything, but he was wearing the faceplate then, so they didn't know.
Why Barbara didn't know anything... "Fear State" and all that was going on with that, I guess? Who knows. And Bruce obviously had a lot to deal with during "Fear State."
Hey, it's pretty dopey to me too. What can I say...
Jace's book, even for the short time it was set in Gotham, was really pretty self-contained. And now that's it's in New York, nothing has really changed. It's a Bat-book onto itself.
Tatsu Yamashiro knows who he is, she briefly trained him.
Or to be more exact, she knows who Jace is, by his name of Tim though (or that's what she calls him or whatever). I don't believe she knows that he dresses as Batman now though.
Either way, again, I can only assume Lucius has talked to Bruce.
"Bruce, you remember my son Tim, and he goes by Jace now? The one who killed someone in a hit and run and that we sent off to military school. Well, he's in New York City now and he dresses like you. So just as an FYI."
And Bruce might be like...
"I see. Well, my Tim just got shot in the neck. The police think I killed Cobblepot. And I believe Selina is sleeping with someone. I have a little bit on my plate right now. But if it's your son and you vouch for him, then okay. I'll look into it when I get a chance."
And that's how the conversation may have gone. Although, Lucius may have not said "FYI." I don't know, I'm just freelancing here.
No, I don't know when they'll deal with the Bruce situation. Maybe next year once this Dark Crisis is over. Or they may never do it, who knows?
I kind of hope they don't, but who knows...
EDIT:
Or if or when they do meet, hopefully they'll have Bruce talk him out of dressing like him (or tell him, either way).
That would be perfect to me. I would be happy.