We already learned that Bruce figured out the basics of Boosters' mission which Dick read from the file.
Dick didn't learn about Booster's secret mission from Bruce's files, this comic is the first time Dick is learning about it.
he chose to tell Dick because he trusted him, he needed a confident
And within the context of this story, that confidant could just have easily been Barbara.
With Babs, well how well does Booster really know her?
He knows that his best friend in the whole world loved and trusted her, and he's encountered her twice in this series during his time travelling, and one of those times involved the most traumatic night of Barbara's life. At this point Booster knows Babs better than he knows Dick.
While we're led to assume him and Dick weren't tight before this there's a level of guilt with her that there wasn't with Dick
Which actually works in their favour, considering both Booster and Babs feel guilty about not being able to prevent Ted's death. Hence the potential for bonding and becoming friends.
I suppose at the end you have to wonder would Bruce have kept it from her for the sake of Boosters' mission or revealed it? Though this is the same Bruce that while knowing what Booster would do treated him like crap before going, "you can stop looking. Blue Beetle is dead. See? Bloody goggles."
Bruce was definitely being a jerk there, but he was also dealing with an Orwellian overlord partly of his own making taking over the world and waging war on metahumans.
Besides which, Bruce is closer to Barbara than he ever has been to Booster, and once it was okay to confront Booster about it without disrupting the timeline, there was no reason why he couldn't let Babs know what happened. I like to think that Bruce does occasionally do the right thing when it comes to people he cares about.