Welcome to the world of owner surrenders.
Now y'all know why so many rescues do NOT fool with owner surrenders.
Pulling from shelters can be a bit of paperwork and logistics arrangements, of course, but accepting animals from the KuhRayZee public is a whole other ball o' wax.
The best thing you can do, OP, is be very, very unemotional with this person, set reasonable boundaries in a polite manner--and stick to them--and make sure she knows by your attitude you are willing to walk away from the situation.
That sounds harsh, but really, a lot of this is like hostage negotiation. I've long felt rescue volunteers could benefit from that training, btw, not just police officers and the like
And, you're right, no matter how much you see it, it never ceases to amaze the way folks will want YOU to pay THEM in order to solve THEIR problem. (That they 'most always created, in the first place.)
And, I crack up when they talk about "being picky where she goes to," and "finding a forever home." Like their own home was NOT. <---Don't dare criticize them for their subpar care and commitment, though, or deal will be so very off.
Thank you for trying to help this girl, but stay safe (with both two and four leggeds) and try to focus on realistic goals and boundaries.
Wishing you best luck.