While your idea of getting the school system’s secretarial stamp on the documents and getting yourself a copy of the stamped documents seems a better idea, if you were ever pressed in court, and the school system held that they never received the documents, your photocopy wouldn’t necessarily hold up as evidence or even be accepted as evidence by the court.
Sending documentation through certified insured return receipt is probably the safest way possible to make sure you have a solid paper trail for your documentation. I’ve never heard of such not being accepted as evidence in a court of law.
I’m talking through experience with HIPAA insurance compliance, and the guidelines that the insurance companies and the states Insurance Departments/Commissions send out to people dealing with HIPAA paperwork.
My method was to hand-deliver the paperwork, and have a separate paper with the words “Received on behalf of [school district] home-schooling affidavit for [child’s name]”, then asked the school’s secretary put signature, title, and date on the receipt. Also had them sign a receipt for the work portfolio we’re supposed to hand in at end of year