http://h10025.www1.hp.com/ewfrf/wc/document?cc=uk&lc=en&docname=c03489643#N804
The embedded key cannot be used to install Windows 8.1 because it is for Windows 8. Therefore Microsoft will not let you download a Windows 8.1 ISO with an 8 key.
The embedded key will not install a disk based version of Windows 8 (or 8.1). It is only valid for the pre-installed version as supplied by the manufacturer. Therefore Microsoft will not let you download an ISO of Windows 8 (or 8.1) as it would serve no
purpose.
In fact there is no situation where the embedded key can be used for anything which is why they do not bother to put it on a label you can read. Belarc's functions are very informative but knowing the key is valueless.
If you have not created the recovery disks as detailed in the link and the hard disk based recovery does not work again as detailed in the link because you have modified it in some way then you can only apply to HP for a disk set (at a charge) or buy a retail
pack.
When you upgrade to 8.1 via the store it is never activated, it can't because the 8.0 key cannot activate 8.1. The upgrade will only work on a 8.0 system that is already activated so obviates the need to reactivate.