Bryce Wayne, aka The Drowned
Yes, there is a female Batman in the DC multiverse: Bryce Wayne, also known as The Drowned. From Earth-11, she is the counterpart of Bruce Wayne / Batman from Prime Earth. She first appeared in the Dark Nights: Metal event (2017), where she became part of a sinister team of corrupted Batmen from the Dark Multiverse, known as the Dark Knights. Bryce Wayne was the sole woman among the Dark Knights, who were all the Bruce Wayne / Batman of their respective universes.

To address the comment asking if she goes by “Batman”, I’ve read the Dark Nights: Metal (Volume 1) series and the one-shot Batman: The Drowned, which reveals Bryce Wayne’s backstory, and as far as I can tell, she was not called Batman (nor Batwoman). That makes sense, since all of the Dark Knights are the “Batman” in their respective universes, and it would be confusing to call her “Batman” in the story. She is usually called “Bryce”, since she’s the only one with that name, while the rest of the Dark Knights are all “Bruce”.
But I think it’s fair to say that she’s a Batman, just like the rest of the Dark Knights, because the issue that tells her story, Batman: The Drowned, is a “Batman” title, not a “Batwoman” title. And as a member of the Dark Knights, she’s grouped with other Batmen, not Batwomen. She’s also obviously a gender-swapped version of Bruce Wayne (the Batman), with a similar origin story of how she became Batman.
I believe Bryce Wayne is a valid answer to what the OP is looking for: “a woman [who] put on the Batsuit made by Bruce [or Bryce] Wayne and claimed the identity of BATMAN” and "otherwise posing as Batman, rather than the obvious answer of Batwoman or Batgirl."