King really doesn't understand Booster

SPOILERS AHEAD

We have attempts at humor that just fall flat especially given the tone. I question why Booster was even in this since his story seems pointless. He's traumatized for still unknown reasons, he doesn't get any help and he's treated like crap.

Remember how Booster almost killed himself trying to save Barbara from being paralyzed ? Or her history with Ted ? Seeing her punch Ted to let Booster be vulnerable when Harley is trying to kill him really makes her behavior worse, right?

Let me get this straight: Rip Hunter is not only in contact with Batman, he's also supposedly  working with him and Bruce tells Barry about the connection. Does anyone else feel like this happened because someone at DC told King that Booster works with Rip ? I don't think he ever explained why Booster doesn't time travel in this so maybe this was a hasty excuse. But it makes no sense.

First Batman telling Barry about this when in a previous issue he didn't even believe Booster was the killer? Why would he do that? He knows Barry blamed Booster and he's betraying Boosters' trust giving out that information.

Why would Rip give Batman ANY information that has to do with his work? He was reluctant to allow John Stewart to have a GL ring with future information and only did so because he knew John wouldn't look anything up. Batman is trying to solve a murder and he will either find out the truth or have access to something he doesn't intend to use which makes no sense.

Rip probably knows who the killer is so Batman not getting that information just raises so many questions. Hell Rip could have said "Booster isn't the killer." It didn't need to be mentioned, all Bruce had to say was that he checked all of Blue and Golds' hang outs.

Rip doesn't need to talk to Batman if he doesn't want to and could EASILY avoid him. There's no reason for him to work with the heroes. We know he's Boosters' son (I don't think King knows though) so why would he help? We also know he's not actually doing anything to find Booster himself since Rip could have found him with no difficulty due to the time tech.