Honestly, I always found this scene a bit... weird. Clark and Babs aren't precisely friends or anything, but they're definitely allies. If I'm Lois, I hear about this "evil hacker Oracle" who needs to be exposed, I do some digging on Katarina and Babs, I hit the Justice League connection, and I immediately turn to Clark and go "hey, you have a coworker named Oracle?"
Clark says "sure, yeah, she's one of Bruce's. Great with research, good friend of Canary. Why do you ask?"
And that's the end of it.
I mean, Lois is not going to go after one of the people who helps keep her husband safe. And Babs would be perfectly aware of that. So it's not an especially plausible scene to me as a reader - Lois is not a bad guy, and even if Barbara's fear and backpedaling made any sense to me, I wouldn't be able to sympathize with it in the least, because regardless of what she might think, it's obvious to me she's not in the least bit of danger here. And Lois tweaking Babs and messing with her head doesn't make much sense to me either - I can see her bringing up the evidence against Babs just to let her know what trails she needs to cover better, but I'd expect her to make her distaste for Katirina clear right out of the gate. As is, she's basically just being an inexplicable ass here to someone she knows is a white hat.
I mean, maybe she's got issues with some of Barbara's methods or missions, enough to feel she deserves to be messed with a little, but there's nothing in the actual conversation to suggest that, and it still wouldn't explain Barbara's freakout or give me the reader any reason to think Lois is a threat.