What do you mean by "serious"?
I know which scene you're talking about. To me that's a girl calling a guy on his BS. That relationship could be best friend or kid sister, or any number of things in between. It's not long before M'ghann and Bee are joining in, either.
The way I see the YJ Barbra is basically the only girl in a family of boys. There's Dick, who she has competed and compared herself against from the beginning. Jason, who she was likely involved with his training and whom she acted like an older sister to (at least if it was anywhere near the comics). Tim, who she's obviously got a good relationship with but is also an older sister figure. And Bruce. That's a pretty tough family to fit into and she likely learned early on that she couldn't put up with any of their BS.
t. That's my 2 cents on that scene.
I would define serious as a committed, long-term relationship. Dick is willing to do that -- he very openly loves Barbara. But it hasn't happened yet. However, that doesn't preclude them having intense petting sessions.
I don't agree that Batgirl would call Nightwing out like that just to call him out on his BS. Nightwing isn't sexist, and the women on the team know that. He might have been a little protective of Barbara in the beginning, but that was probably a hurdle that they already crossed. I do not think that this happened because Barb was a girl, but because she was a rookie and, in some part, under his training. I ultimately feel that the girls really got a kick out of flustering the usually cool and collected Nightwing, and that it added a moment of levity to a very nerve racking situation.
I think that the only scene that actually shows any sort of relationship between them happens in Downtime, I think? When Dick takes a picture of himself with Artemis at Gotham Academy, Babs seems to me to get a little jealous.
Babs: "Who is that, Dick?"
Dick: "Just a friend, Barbara."
I disagree that the fact that she is a female would make things at the Batcave harder. This Batman is very open minded, and he doesn't display the same prejudices that he usually displays in other media, and if anything, I feel that things would be easier for her since she has Dick to explain things. Bruce can be brusque and short, but Dick has developed his burgeoning team skills by this time. Besides that,
Dick already understands Batman, and gets what his methods are trying to achieve.
Lastly, I also feel that Dick, post Aqualad's assumption of leadership, has stopped being jealous and openly competitive. He is a mature team player and a very competent leader. He would have wanted for Barbara to succeed, and would have been driven to make her better than he is.