To provide an update on my issue, I did a fresh install of Windows 8 and then ran critical Windows Updates. After doing this, the Windows 8.1 update option showed in the Store and I went through it, without a single hitch this time.
Goes to show that this isn't a hardware problem, but a software issue that appears to be driver related per everyone's posts.
The ridiculous part is that I wanted to do a fresh install anyway and all this hassle would've been avoided for me if only Microsoft had made it possible to download Windows 8.1 as a standalone ISO with my perfectly valid, perfectly legal Windows 8 key instead of pulling this shenanigans.