@ -midnight
Normally your product key is:
1. On a label stuck to your system.
2. If you got a CD/DVD copy of the OS with the system then it could/should be on that.
Outside of that perhaps Dell can help out with information based on your purchase.
I don't know why you should be forced down the route to purchase Windows 10 Pro. The real problem, issue is activating Windows 10 Home.
Have you tried the Troubleshooting option and if so what were the results ?