In Lucy's defense, she didn't actually demand that James tell her who Supergirl was. She just felt left out because James was confiding more in Supergirl than in his own girlfriend, and it was James himself who had the idea that telling Lucy the truth would fix things. We've seen before that Lucy feels that she's not good enough to compete with a superhero for James's attention. She's insecure about her worth to him, and he wanted to assure her of her worth by showing her that he trusted her with his greatest secret.
Although if he had told her, it probably would've backfired. Knowing that James worked with Supergirl on a daily basis at the office as well as in crimefighting situations would've just made her feel more marginalized.
Now that I think about it, I feel sorry for Lucy. What has she got to be so insecure about? She's staggeringly gorgeous, she's smart, she's likeable, she's a successful professional -- she's someone who makes other women insecure. But evidently she can't see that, and that suggests something rather sad about her upbringing. We've seen what a jerk her father is -- she's probably grown up being constantly belittled by him, made to feel she's not good enough. I wonder if Lois is the same way in this reality.