I was in this boat (similar) a while back. Bought a refurb Win 7 PC from newegg. Came preinstalled with 7 home, and a refurb tag / key for ultimate. The only solutions offered were to return the PC for the fix, or shell out money for the proper copy of windows to match the key. I was fine with running home, if they would have given me a key for it...but they would not.
After some digging, I found a way to access the no longer available links on the Microsoft, to download the ISO that matched my key, and was able to install it that way. Microsoft stopped allowing access to pre-Windows 10 versions ISO downloads, even for those who have valid credentials. So after finding a workaround, I did a backup of everything, reinstalled the correct ISO and had no issues keeping all of my data. If you have questions, feel free to PM me.