I don't think there are any anymore. There used to be a few. I looked pretty hard in 2015 because I had the same thing, early October birthday girl who is in some ways advanced.
You could try the community-based organizations, I'm not sure how it works but maybe you could go for free after her birthday.
https://www.myschooldc.org/find-schools/school-options-outside-my-school-dc
I'm sure you'll hear this from a lot of people, but it turned out that my daughter's age-rule placement was the right place for her socially. While she seemed to have good social skills in terms of speech and being generally a calm and cooperative kid, her social skills in upper elementary were lagging and she would have been miserable in the grade above. It can be such a good thing to be one of the older kids. And looking back, it seemed like a lot of money but it wasn't worth giving up an entire year of her childhood at home with us. Life comes at you fast!