Issue seems to have resolved itself. I didn't really try much since the last time. I installed Wireshark on the 3CX server, and then went into the 3CX Phone System Updates -> 3CX Service Packs/Updates. I captured the network packets while doing so, and after locating some HTTP Get Requests for some XML files on a server on the Internet, I tried to manually browse to the site.
A few minutes later, I got an e-mail that a service pack was ready for my server, and then lo and behold, the 3CX Service Packs/Updates turned red. I went back in there, and SP3 was now available.
I still have no idea how or why it fixed itself, but I'll just accept that it's now installed, and move on. Thanks for all that replied.