Missing digits from Dell product key

I had to have a new motherboard istalled in my pc and I need to reactivate Windows. My issue is because of scratchs I can’t read all of the digits of the product key which is on a sticker on the side of the computer. I called dell and they said that they can not retreive the number because my warrantee is expired and will be charged $49. for tech support to get a new one. They can identify my by the express service code I gave, but the can’t find the product key #?