“Extremely quiet” isn’t really a character flaw; just characterisation. Perhaps ‘timid’ or even ‘mousey’ (for the animal connection — she’s gonna have to find her ‘roar’) work better to describe the character?

Is the story really about negotiation? Does it mostly take place in an office (or circus equivalent … a trailer, I guess?) with the main character arguing the specifics of the trade with the owners of the tiger? I suspect it’s more a story of re-habilitating the damaged tiger (and in so doing, finding her own courage) — right? If it really is about the negotiation, then I guess the logline is accurate, but the premise isn’t particularly compelling. If it’s about something beyond the negotiation, then the logline needs to be reworked to focus on that.

It might help if there is something specific she needs the tiger for? What I mean is that; if a tiger died because of her incompetence, it wouldn’t be up to her to replace the tiger. The zookeeper would be fired for that incompetence, and presumably someone who runs the zoo would be in charge of sourcing a new animal; if they bothered to at all. However — if it wasn’t her fault, why would her job be tied so closely to the tiger exhibit? Why would she be OUT of a job — wouldn’t it just be part of her job to find a replacement? The stakes aren’t very clear to me.
So the point I was getting at was — maybe a specific reason she needs the tiger. Maybe?the zoo?were planning on breeding the tigers and they only have a small window of fertility with the female tiger, so she’s got to find a replacement for the dead male? Or maybe there’s someone coming to the zoo in a few months to see the tigers (though outside of a royal, I can’t imagine why someone’s job would be in the balance over it), or maybe the zoo is about to close and they wanted to sell the tiger to keep it open, or else the tiger was the only reason anyone was coming to the zoo …

I guess, as I write my thoughts on your logline, it becomes apparent to me that I think your story has a stakes problem — I?don’t understand why the protagonist’s job would be in jeopardy over this situation, and I think the premise needs to be reworked.
Likewise — all of the above, what is the action the character takes to achieve their goal.