We tried a different brand, same problem.
The server is running headless, so we connected a monitor (if only to be able to troubleshoot this more quickly) and oddly, that seems to have fixed the problem (i.e. after logging onto the console).
Now, even if we disconnect the monitor again, we can remote desktop to the server and see drive letters every time we change the drives. For each drive, we removed the default partition and created a new NTFS one (we tried this previously with one drive and it didn't work), plus removed the letter assigned to the virtual CD-ROM with the password software, and the settings are now preserved when disks are changed.
I'm at a loss, but am pleased it's working now!