How far are you willing to travel ?
I've heard good things about Basis. Have you explored other areas such as Palo Alto (it has an excellent school district) ? My child goes to public school, and, while it is true that they can only do so much, if you are in a school district that is responsive, they are a bit (not a lot, but better than private schools who won't do much) more flexible and will do things like pre-test, accelerate the curriculum etc.

Palo Alto, Cupertino, Piedmont and San Ramon school districts are generally supposed to be good. I don't think a gifted program exists till middle school for some of these school districts. But I also do know that they are willing to help/differentiate, especially if they also view the child as gifted and see that the child's parents are very involved. My child has 2e issues, and is still doing okay. If your child has no 2e issues, and is gifted, I think some public schools will be a good fit. Also, I am not sure about San Francisco, but every third kid in the South Bay, in well-performing school districts, seems to be gifted. The only difference is in the LOG.

Hope this helps ! Good luck !