The Villainess might work. I see it's on your Plan to Watch, and you've given favorable ratings to some other dark, gritty films and shows. The female lead isn't a psychopath, but her experiences are harsh.
Japanese horror film Audition is high quality, and the female protagonist should satisfy.
I only got a couple episodes into Thai drama Girl from Nowhere - it was too disturbing for me at the time. My take on the female protagonist from my limited watch is that she's a remorseless, avenging angel type.
Korean film The Call has a female antagonist and a female protagonist that are the two main characters. The antagonist fits your requirements.
It's going to sound a little off the wall, but I wonder if you'd like the classic Japanese art film Woman in the Dunes. The female protagonist is anomalous.
The female protagonist of kdrama Family Matters isn't a villain, nor is she a psychopath, but she's morally ambiguous, and you don't get the full story about her identity until the end of the drama.