Just an FYI, I noticed another person in the past few hours posting the same problem, and my behavior is the same as his.
If I turn off the "Updates" classification, the synchronization will work fine. Turn it back on, and it fails immediately upon starting.
Here is his thread for reference:
https://social.technet.microsoft.com/Forums/en-US/7c3e5c52-13b2-49db-a631-19e23ac7763c/wsus-2012r2-not-sycing?forum=winserverwsus
At this point I'm going to assume it is an MS issue until I get more information, because I can't burn too much more time on this right now.