Just to clarify this for anyone else that's reading:
The client-side AME is installed automatically with CS6 applications. It is this component that is failing to validate my CC subscription, causing the application to fall back to requesting a serial number.
After a telephone call with Abhijit, we discovered that installing the standalone CC desktop application, either before or after the CS6 package, allowed the subscription to be validated and the software activated. It seems like there's a significant bug there, or at least a lack of consideration for the possibility that clients might not have CCD installed. The workaround is just about satisfactory for my environment, but I think this could still cause trouble in some larger or educational setups.
A useful tip: The CCD installer downloadable here: https://ccmdls.adobe.com/AdobeProducts/KCCC/1/win32/CreativeCloudSet-U p.exe supports the same command line switches as msiexec, so although it's not readily deployable as an MSI, it does allow for unattended installation.