It seems that there's an additional folder inside the container that Tachidesk will download images to temporarily.
Under "/tmp/Tachidesk" there's a "manga-cache" folder as well as a "thumbnails" folder. The "thumbnails" folder seems to be used (expectedly) for image thumbnails while you're browsing comics. "manga-cache" seems to be used as a temporary cache while reading/downloading comics.
I decided to go ahead and map "/tmp/Tachidesk" as an additional Docker volume. This has two advantages:
It avoids bloating my Docker virtual image with a bunch of temp files
It makes it slightly easier to inspect what's happening on-disk when Tachidesk has trouble downloading a comic
Not sure if those use-cases common enough to be worth including these paths in the default template, but I thought I'd drop a note in case it helps anyone else who's trying to figure out where these temp files are ending up.