After double-checking file system permissions, and running Filemon to try and figure out where stuff was failing, I found out that the task scheduler actually creates a log file. By default it seems to live at %SystemRoot%\Tasks\SchedLgU.Txt, but that location can be changed in the registry by modifying the LogPath value at HKLM\SOFTWARE\Microsoft\SchedulingAgent.

That log file contained an error message (well, several...) that quickly lead me to Microsoft Knowledgebase article KB867466. Turns out that on Windows Server 2003 member servers, non-admin users running non-interactively don't have Read or Execute permissions to cmd.exe...

After fixing the permissions, the job ran fine.

That's when I discovered that the batch file was making some assumptions w.r.t. the user's regional settings, so I switched to a VBScript instead. If I'd done that in the first place, I wouldn't have had to spend 15 minutes fighting with the task scheduler, since the permissions issue doesn't apply to cscript.exe...

Oh well.