Solved! Despite chowning the /docker/qbittorrent directory (and below) to the UID/GID I am using, that user was denied access to the parent directory within Synology’s Disk Station Manager. I’m not even sure how that gets implemented at the filesystem level, but apparently it was enough to prevent Docker from creating the .cache directory despite the fact that it could happily create the other directories. After granting the user r/w access via DSM, it’s all working perfectly.
Sorry for the false alarm. Hopefully my experience can save another Synology user some time in the future.
Cheers!