We called the local public and private schools (basically anonymously) to gather information and got pretty much shut down by the public schools and one of the private schools. "No, we do not do that, goodbye."
Another parent told us that what we would need to do if we wanted to push harder on the public schools is to request a very vague meeting in person with the principal (she uses, "We're trying to decide where to move") and bring it up face-to-face after they've already invested some time in it.
We're doing private for the next year or two, but we're keeping that in the back of our mind in case we need it.