There's a precedent for early entrance to first in NH in spite of district policies like the one you describe: see this web page.

Citing precedent to them, if it comes to it, is a good way to help win. Especially if it takes them a couple weeks to notice he's been enrolled in 1st and he's been doing fine. It's easier to get forgiveness than permission....

If you discover that he's enrolled in first grade and decide to chance sending him, you might want to keep your head down. If they do figure it out, you can respond that you called and asked. When you were told he was enrolled in 1st, you assumed they changed their minds.

Alternatively, would Montessori enroll him in 1st? If he did that, the public school would probably have to enroll him in second after that.

Tough call!

Val