So, just saw the movie last night, and ya'll mothers are freaking out over nothing.
I was kinda thrown off with the inclusion of the sexual relationship between Bruce and Barbara, because like many, I always saw it more of a mentor/mentee relationship, but in the movie she is a grown, adult woman and as a partner-in-crime(fighting) to Batman. She doesn't come across as "pining" for Batman, nor is it ever insinuated that the only reason she is fighting crime is to get Batman's attention. Anyone saying that is pulling shit out of their ass to stir up controversy, or maybe projecting

In fact, when her co-worker is talking to her about her frustration with her "yoga instructor" and recommends that she do yoga somewhere else, she's frustrated that she can't. She's alluding to fighting crime elsewhere, not chasing Batman. She would stop being Batman's partner, but can't because she can't just leave Gotham.
The actual scene itself isn't the best done, but not the worst. Its the cliche mix of violence+sex as the two of them get frustrated with each other, cut lose fighting and end up fucking. Nothing is really seen as it cuts away pretty quick.
Afterwards, Batman is shown as the one who is acting weird over it. Barbara is frustrated that he is avoiding her, and calls him up trying to get him to relax saying, "It was just sex, it doesn't have to mean anything." so if anything, she seems more emotionally stable/stronger than him in that aspect. Which, while Batman is known to have flings as Bruce Wayne to keep up playboy'ish appearances, he isn't known for being very emotionally stable with relationships.
I found it mostly to be a bit of padding to flesh out the runtime of the story, as well as build up the character of Barbara/Batgirl for the audience to make the impact of the Joker's later actions more impactful. This is why:
Joker doesn't know who she really is. He only knows her as Commisioner Gordon's daughter.
Joker doesn't know she is romantically linked to the Batman.
Joker's ambush pretty much comes out of nowhere (unless you have previously read the book, of course). The scene goes from casual, Daddy-Daughter moment to "oh shit!" fast.
So, while Barbara is an important character to the audience,
she is meaningless to the Joker. She's just a way to "get to" Gordon, mentally and emotionally, to the Joker. The fact that he shoots her so nonchalantly is meant to shock the audience. Which, even though I knew the story well, when she was opening the door and you see the Joker's grin, I was still cringing a bit internally, knowing what was coming.
The first bit of story actually builds up the character so she is an active participant in the story, and more meaningful to the audience, instead of just being a bystander who gets shot as part of the Joker's scheme.
It works, for the most part.