Still ultimately she's being threatened by Spy Smasher here. That's who's feeding Lois info IIRC. SS is a bit player in the world scheme. Bruce Wayne (outside of Batman) is one of the richest, most connected, most powerful men on the planet. Superman is revered as a god by some people and (since this is prior to all this Sam Lane nonsense) has the ear of Washington (and his best boyhood friend is the former President of the US). Wonder Woman is the most admired woman and listened to woman in the world (even after the Max Lord stuff). Babs' own best friend is one of the founders of the JLA. Heck, all Bruce would have to do is place a phone call to Sasha Bordeaux (of Checkmate) or Lorraine Reilly/FireHawk (who is also a US Senator from NY in the DCU) and Spy Smasher's threats (and her personal/professional life) will start to fail quickly. All Babs would have to do is ask Bruce to talk to Lois or Dick or Dinah to talk to Superman (who would then talk to Lois).
Instead Babs thinks trying to threaten to expose the wife of the most powerful and most popular man on the planet is a good idea? In any reality? She's supposed to be one of the smartest people in the DCU - she knows Batman and Superman are BFFs. Being who she is she probably knows Lois and Bruce are tight - to the extent he gave the Kents their home and they live in a Wayne Building (and Oracle has probably done her research on LL prior to this meeting so she should already know that). If Lois does know about Oracle, she would also know through her connection to Clark and Bruce the connection between Oracle and Batman. And Lois would protect Batman over a story (not to mention the fact that when this story came out I believe Bruce still owned the Daily Planet). If Babs is using that big brain of hers, she should be connecting the dots here and the fact that Lois knows a lot she has never written about (heck, just a few months later Lois was one of the few civilian attendees at Dinah's Bachelorette Party/Shower with Babs in attendance as well)
If she's exposed by Lois or Spy Smasher, the super-hero community would 100% circle the wagons to protect her, Bruce would make sure of it. If however she exposes Superman, even Black Canary would wash her hand of her - and then she'd have no friends at all.