If you hear a rumor that someone is your husband's coworker, asking him "hey, is this your coworker" is not relying on him to do your job. It's common sense. Lois is not going to burn sixteen hours chasing paper trails and rumor to try to figure out if Oracle is JLA when she can spend five seconds asking Clark. Wasting time and effort is both stupid and bad journalism. Using available inside resources is smart and good journalism. Are you going to tell me Clark is somehow "doing Lois' job for her" when he gives her a Superman interview over dinner? If she asks Diana about the Themysciran government, is Diana "doing Lois' job for her"? If she calls up Bruce Wayne to ask him about a Waynecorp merger, is Bruce "doing her job for her"? If she needs a piece of information about the Justice League... the first thing she's going to do ask a Justice Leaguer. That's how journalism works.
And where are you getting this "ask his okay" nonsense? How is saying "hey, Clark, is there a JLAer named Oracle" somehow "asking his okay" to pursue a story? She is pursuing the story by the very act of asking him. Superman is a knowledgeable source who confirms the rumors that Oracle is a good guy. Boom, end of story. That's not Lois dropping her pursuit, that's Lois finishing her pursuit.
Besides, even if she doesn't ask Clark, and, like, breaks into Firestorm's laptop to check out the League roster or some similarly pointless, convoluted and unnecessary but-hey-at-least-she-did-it-herself nonsense to confirm Oracle's League membership, the point stands that she walked into that restaurant knowing "this is a woman who helps keep the world safe," and Lois is too much a white hat to be a douchebag to a hero for no reason. Which Babs should know, because she knows Clark wouldn't marry a douchebag. And which the reader does know, because if they know nothing else about Lois, they know Superman wouldn't marry a bad guy. And only a bad guy would go after Oracle. "Superman's wife puts Barbara's life at risk" does not compute, it's really that simple, and that has nothing to do with valuing Lois' relationship to Clark over anything else about her, and everything to do with the fact that that's a fact we know Barbara has and a fact the reader is guaranteed to have and therefore an equation both will be running and both should be turning up as an error.