The filename is set on the initial upload, then when syncing, the same filename is kept. This is for safety reason, WordPress doesn’t handle natural file renaming and this is a dangerous process, and I don’t want WP/LR Sync to include a tricky/dangerous option. Why dangerous? Because in WordPress, once a Media is alive, there are many relationships between this Media and other parts of WordPress, and that depends also on plugins. Changing this filename would break all those relationships.
I actually developed another plugin called Media File Renamer. This plugin does renaming, so I am trying my best with that plugin to handle most issues (= relationships). Please have a look at it ?? It doesn’t rename from the file set in Lightroom but it’s using the title of the Media.
I’ve tried this but it doesn’t appear to work – in that case do you need to remove the old image file from WP and republish the image from LR again with the new filename?
You could use Media File Renamer or you can delete the Media, Publish, re-add it, Publish. If it’s a rare operation and that you really need it, then it’s best to do this way.