Have you been to the HP support link for upgrade from Windows 8 to Windows 8.1? The link is:
http://www8.hp.com/us/en/ad/windows-8/upgrade.html?jumpid=reg\_r1002\_usen\_c-001\_title\_r0001
You might help you transition to 8.1. I have an HP ProBook 655 G1, and when I upgraded to 8.1 it did not invalidate my serial key. This link should aid in your transition. Also it detects what HP product you are using, and should give you what you need, specifically for that device. My key didn't change until I downgraded to Windows 10. It did stay activated from 8.1 to all Insider Preview builds of Windows 10 evaluation. I went ahead and installed the Commercial version awhile after release, and then my key was changed and activated. I still continue to get the insider builds, as I opted to continue in the program. We are on build 10547, with no indication that a final build will emerge any time soon. Give that link a try and see if it helps. If you continue to have issues, reply back and I will see if I can find a better solution. As you have experienced, assistance from Microsoft is an oxymoron. HP is in sort of a love-hate relationship with Microsoft. So, HP prefers that end users go to them on these types of issue. Either way, please let me know it this works out.
Michael
Black Hat hacker working for the common good
Microsoft. Come on, admit it. Your life would be dull without us.