.vw-content-list-watched gets its style rules from XenForo's @DicussionListIcon style property (see vw_content_list.css), which in the default style sets up the xenforo-ui-sprite.png inclusion, but this is modified in Flat Awesome. It may be possible to detect a Flat Awesome style using the unique style properties that it has, such as fa_version, and to include changes in the default style, but for large numbers of custom styles, this could make the core template code become unruly and bloated for people who don't even use some styles.
This is not a bug in VaultWiki directly. In the future we may release a Flat Awesome style for VaultWiki templates, but as an add-on.
Until then, please make your desired custom edits to your style using VaultWiki's vw_additional.css template.