I wouldn't do it. I had one in private and one in public for high school and the public school experience (in MoCo) was just as good as the private school experience. If you need to move for a better public school it may be worth doing it.
You will get financial aid, but as others have noted most schools expect families to still pay a good chunk of tuition so you could still be on the hook for $20-30k/year. Or if they can't meet his need they won't admit him. That happened to a friend's kid at GDS - she was put on the WL because they didn't have enough FA.