I do download the stories as EPUB, the tab setting is enabled and the cover doesn't show up in Calibre even before I convert it to MOBI (even if I download an entirely new story).
Here's a
screenshot from a
random story I downloaded, with the configuration I posted earlier. As you can see the first page is the cover, but is not displayed as the cover image in Calibre or on my Kindle Paperwhite after conversion.
EDIT: It really was a random story, I don't read Naruto/Sasuke slash stories...
.