Update:
I never heard back from the school, but I had to get a medical form filled out and bring it back to the school, so I did that today. The principal is on vacation and the secretary didn't appear to remember our situation, even after I reminded her.
She did check the computer, which says 3rd grade (what we want), so she said he's registered for 3rd as far as she knows. I also have the copy of the registration form, signed by me, which says 3rd grade. She said I could call back the week of August 13, and the principal would be back from vacation. But, I'm inclined to let sleeping dogs lie because if they want to change it to 4th grade, they'd have to get in touch with me. Right?