I suspect the number as indicated in the how to find your serial number documentation is adequate, and that includes the receipt and box. It also mentions an invoice - Viewing & Changing Orders - Shopping Help - Apple - https://www.apple.com/shop/help/viewing_changing_orders - "Once an item is shipped or picked up, Apple will create an invoice for your order.
Replacement instructions for Airpods are provided at the bottom of this support article "How to find your lost AirPods, AirPods Pro, or AirPods Max": --> https://support.apple.com/HT207581 Neither the warranty nor AppleCare+ helps with the cost of Airpods that are lost or stolen. The cost for damaged Airpods may be lessened by any 'AppleCare+ For Headphones' coverage you may have.
'AirPods Service and Repair' lists replacement fees. Click here --> https://support.apple.com/airpods/repair/service
1. If you can't find your AirPod or Charging Case, or if something is damaged, get the serial number. Click on this link to discover how to find the serial number. --> https://support.apple.com/HT209371
2. Contact Apple Support for a replacement. Click here --> https://getsupport.apple.com/?caller=kbase&PGF=PGF32003&category_id=SC0298
(You may have to go to the support article in my first link for the special Airpod replacement service link if that one doesn't work.)
3. Learn how to set up your replacement.
Set up your replacement AirPods or charging case - "When you get your replacement, follow these steps to use your AirPods again." --> https://support.apple.com/HT207827