Well, perhaps in the fanfic, Harley Quinn has kidnapped someone important, like a scientist who was working on a new technology that could be dangerous in the wrong hands. Batgirl hears about it and goes to confront Harley. There's a series of battles where Harley uses her acrobatic skills and her signature mallet, while Batgirl uses her gadgets and martial arts. In the end, Batgirl manages to rescue the hostage and send Harley back to Arkham Asylum.
It depends on how the writer wants to portray it. But generally, Batgirl has a good chance of winning. She has extensive combat training and is highly intelligent. She can analyze Harley's moves and come up with counter - strategies quickly.
The emancipation of Harley Quinn has been depicted as a comic. It explores her character development and story in that medium. There are various storylines and art styles that contribute to the overall narrative.
Harley Quinn's sexuality in the comics isn't strictly defined as gay. It varies across different story arcs, with her showing interest in characters of various genders.