I recently saw "Gone Girl" (spoiler alert!) and have been reading articles about
I think it's hard to answer this question without going into the details of particular narrative or representational works. It's an important question, but maybe not one for which a decisive philosophical answer is possible.
Let me point to one step in your message. You write of creating "characters that exemplify such stereotypes. Women are sometimes psychotic"; and so on. Now, in the step from the first of these sentences to the next, you show that you are using "stereotypes" as identical with "generalizations." It's true that many general statements one could say about women (or about any other group you choose to think about) are sometimes instantiated. Philosophy professors are sometimes self-obsessed; therefore, someone writing a screenplay about a philosophy professor (not as glamorous as the screenplay to "Gone Girl," I grant you) should be free to make that character self-obsessed.
But these feel like different cases, don't they? I think the reason is that a stereotype is not just a general statement about a particular kind of person; not even a general statement that people frequently say about that kind of person; but rather a general statement that people frequently say, and that is treated as the only thing you need to say about a given kind of person. And sometimes a narrative work gives its characters nothing but stereotypical traits; a single trait is permitted to constitute the entirety of that person.
Look at the husband in "Gone, Girl" by way of contrast. By now our exchange is full of spoilers, so I won't warn the reader of any more. Nick Dunne is treated as a stereotype in some ways. A failed writer teaches creative writing and has affair with a student. And yet he's not stereotypical. He also gets to be a sensitive son; an emotionally communicative brother; a bar proprietor with a sense of humor. Some general statements are true about him that are true of lots of other white forty-year-old men in small American towns; but other traits are specific to him. He has been fleshed out, which means not liberated from all stereotypical descriptions, but rather not reduced to them.
When I think about the woman in this movie, on the other hand, she is close to a one-note personality. (Interesting how the man's twin sister Margo is another kind of one note: no relationship of her own, she is purely a sister to Nick, loving him even when she rebukes him. At least she is a supporting character.) At every turn, Amy schemes and plans. She responds to almost everyone with hostility. Not her mental instability alone, but this presentation of her instability, make for the stereotype.
In short, I don't think writers are constrained by the paradox you describe. They can describe their characters in a lot of ways. But they should be wary of letting those descriptions degenerate into stereotype.