Bryce Wayne, aka The Drowned
Yes, the DC multiverse has featured a female character taking on the mantle of Batman (and not Batwoman). In the Dark Nights: Metal event (2017), one of the primary antagonists was Bryce Wayne, also known as The Drowned. Hailing from Earth-11, she is the twisted counterpart of Bruce Wayne/Batman from Prime Earth.
Bryce was a member of the sinister team known as the Dark Knights, a group of seven corrupted Batmen hailing from the Dark Multiverse. Among these twisted versions of Bruce Wayne, Bryce stood as the sole woman, but make no mistake—she was every bit a "Batman" as her male counterparts.
In the Batman: The Red Death tie-in, The Batman Who Laughs, the leader of this evil ensemble, made it crystal clear that they were all Batmen, Bryce included:
The Batman Who Laughs: And now the seven most dangerous Batmen that never were are here to drag you into the dark.
- Batman: The Red Death
Throughout the Dark Nights: Metal storyline, the Dark Knights were consistently referred to as Batmen, never as Batmen and one Batwoman. Bryce herself was never called "Batwoman" either.
General Sam Lane: Early battlefield reports described the invaders as "Nightmare Batmen."
- Batman: The Merciless
Bryce Wayne's status as a female Bruce Wayne / Batman variant is also made clear in Dark Nights: Metal #2. When Wonder Woman asks where Bruce (Batman) is, Bryce, along with the other Dark Knights, responds, "I'm right here," acknowledging that they are all versions of the same person.
Wonder Woman: Bruce… Where is he?
Barbatos: You want Bruce Wayne?
Barbatos: Look around.
Bryce (along with the other Batmen): I'm right here.
The one-shot issue Batman: The Drowned delves into Bryce's backstory, showcasing how her origin mirrors that of Bruce Wayne's path to becoming Batman in a twisted way. Additionally, the fact that this story is part of the "Batman" line of comics (not “Batwoman”) and that Bryce is grouped with the other Dark Knights (who are all Batmen) reinforces her position as a female embodiment of the Batman identity.
Bryce Wayne undoubtedly qualifies as a woman who donned the Batsuit and claimed the identity of Batman. Her inclusion in Dark Nights: Metal serves as a compelling answer to the OP's question, presenting “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."