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There are two reason WHY the key will not be accepted.
Every Windows software package has an exclusive Product ID. This is part of the 5x5 License key you key in when installing the software and when it is decrypted this is compared with the Product ID of the software you are trying to install held in one of the files on the source. For the same reason you cannot use an Office key to authenticate a Windows install you cannot use a Windows 8 key to install Windows 8.1, they are two different products.
Operating systems supplied preinstalled on computers from larger manufacturers are effectively already authenticated in that they do not have to go on line or use telephone authentication to work (and they have their own Product ID). In order to prevent piracy these keys are tied to the individual manufacturer and the BIOS they have on the PC. Because they are pre-authenticated Microsoft's systems know that no computer should legitimately attempt to authenticate on line or by telephone with this key so any attempt to do so will be rejected.
In order to be able to use HPs version of Windows 8.1 you would need to be able to reprogram the part of the BIOS that is used to verify the validity of the license each time the computer starts. This again for antipiracy reasons is not a part of chip that can be reprogramed during a BIOS update. As a part of the licensing agreement between HP and Microsoft HP can only supply operating system recovery disks to somebody who has proved that their computer had the relevant operating system when supplied.
If Microsoft (or HP) were to supply you with an 8.1 key then effectively you have two keys, that which was part of the supplied PC (8) and the new one. You would also need an image of 8.1 to be able to use the key. As this cannot use the BIOS lock system (because it is for Windows 8) to prevent piracy there would be nothing to stop you installing the second key on another computer.
It is a completely separate issue as to why the Pro Pack upgrade failed and left the computer is a state that even the 8.1 upgrade would not work. As you and HP do not know why there is no direct evidence to say if the same thing were to be attempted you would get the same result. Reading threads on here and elsewhere HP definitely seem to have a problem with the Intel HD driver on their systems under Windows 8.1 but they are not going to admit that.
Ultimately the real problem is the level of piracy protection that Microsoft are having to implement to prevent it happening. They know that in the past they have lost millions of dollars through piracy, it is more than a little unfortunate that the system adopted has resulted in such problems. Coupled with that Windows 8 being such a disaster that they had to roll out a free upgrade and not just an update but you could image the reaction if it had been chargeable. There is no doubt that this could have been handled better but I suspect you know that behind all the customer service platitudes there is legal representative resisting any admission of a mistake and an accountant worried about the margins.