Thoughts on private middle schools in Brooklyn for a very math advanced kiddo

We've been in the public school system since the start and have had great success due to how integrated our school is in the neighborhood. This is in spite of our son being extremely advanced in Math and above average in other subjects. I've been able to augment his education at home and school so thus far we've avoided the pitfall of having him bored in class or resentful of lessons that he feels are too easy. All that being said we're really aiming to pivot into middle school. We've been looking hard at all the public school options but have much less first hand info on the private school offerings.

We aren't looking for a traditional ivy league feeder thats all about rigor and homework. Our son loves to learn and is relatively self directed, so we feel a school with a focus on exploration over competition is ideal. The problem is I don't know if schools like that will be flexible enough to let him push ahead in Math so he's not bored.

I've been eyeing Packer, Berkeley Carroll, Brooklyn Friends, Saint Annes and others. Tuition at these places is eye opening but I would feel remiss if we didn't consider all options. So if anyone has any insider info I would love to hear it. Most of them seem to offer the usual private school plusses, slightly more advanced curriculum, smaller class sizes and so on but its very hard to get a feel for whether a sixth grader who is ready to explore algebra or other fun math stuff would be allowed/encouraged to.