Meanwhile, I like the "natural" tension of Kara being confronted on the DEO ethics, and that the hero, on occasional can stray & needs to be reminded of what's right. So good for Jimmy on that...being a real friend to Kara/Supergirl.
But then we have the opposite with the "I have to tell Lucy the truth"....yeah, that all makes no sense to me either. Hopefully, they can clarify/create a reasonable solution.
I don't see the opposition. In both respects, James's concern is for fairness and treating people right. And it's not like he went behind Kara's back to tell Lucy. He explicitly said that he needed Kara's permission to tell Lucy the truth. He won't betray Kara's confidence, but he's having to betray Lucy's trust in order to preserve Kara's confidence, and that puts him in an untenable position.
Besides, I don't see that big a problem with James's request. Kara hasn't exactly been all that cautious with her secret identity. She blurted it out to Winn at the first opportunity, she made no attempt to hide it from her family, and the entire DEO staff seems to be aware of it. Heck, even Max Lord knows. She kept it from Cat because she was afraid of the media exposure and of losing the job she values, but is there really a good reason not to bring Lucy into their confidence? She's Kara's friend and James's girlfriend, and she's helped the team out on several occasions. And she's a lawyer, so keeping confidences is something she's good at. The one iffy matter is that her father is Sam Lane, but the same goes for Lois, and if James knows that Superman is Clark Kent, then Lois probably does by this point as well. (I'm pretty sure General Lane doesn't know Supergirl's civilian ID. Hank has made a point of calling her "Supergirl" rather than "Miss Danvers" when Lane was around.)