Well to nitpick Bruce already knew and what Booster did reveal--because denial was futile--wasn't much, We already learned that Bruce figured out the basics of Boosters' mission which Dick read from the file. My understanding of this reveal (which was mostly off panel) was that he chose to tell Dick because he trusted him, he needed a confident, and he figured it was easier than doing the time wrap 100 times to get rid of the photos.
With Babs, well how well does Booster really know her? As far as I know they only talked once on the phone before he tried to save her in the KJ. 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. He failed her and neither of them saved Ted. Would Bruce have told her beforehand? He doesn't know but from that POV he'd likely assume that if anyone would have told her it would have been Bruce.
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."