It's the best card Babs has to play, though. She's got no way to discredit Lois on such short notice, and no way to guarantee she's destroyed all of Lois' evidence if she tries. So she really can't protect herself against Lois running the story, and her only recourse is to convince Lois to choose not to run it. And she is a Bat, after all, she's hardly going to appeal to Lois' better nature. :P
And I'm not sure how relevant it is to bring in "oh but all her friends would be pissed at her," anyway - I mean, it's not like Clark would be particularly pleased with Lois if she destroyed a fellow Leaguer, and Babs may not have more friends than Clark in the cape community, but she certainly has a hell of a lot more friends than Lois, so she's facing basically the exact same sanctions that Babs is. It's not like Dinah's going to go "hey, Lois destroyed your life, but whatever, my loyalty to Superman means I'll overlook that and just be pissed off at you." And Bruce would probably already be prepping retaliation of his own.