I've fixed it!
My problem was as simple as the "P" in "Precise" in /etc/lsb-release under "DISTRIB_CODENAME" being capitalized instead of lowercase. It should be:
Other files of note:
As mentioned previously, it seems that one needs to both enable the partner repository and add the lines mentioned to /usr/share/software-center/softwarecenter/backend/channel.py (or uncomment them if they're already present).
Thanks for your input, everyone! I can't believe it was such a silly mistake... ;-D