After running PowerCfg /q and reviewing this page, I believe I have determined the solution to this problem.

There appear to be many power settings that just don't show up in my advanced power options window. One is Sleep → System unattended sleep timeout. To make it visible, I opened regedit.exe and found this key:

HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\Power\PowerSettings\238C9FA8-0AAD-41ED-83F4-97BE242C8F20\7bc4a2f9-d8fc-4469-b07b-33eb785aaca0

Set its Attributes DWORD value to 2. This will make it show up in your advanced settings. Go there and configure it to be zero minutes if you don't want it to sleep when you lock your screen.