Render a file in a specific view mode
You'll want to use the file_view_file() function provided by the File entity module. The function has three parameters: a file object, the name of a view mode (or an array of custom display settings) and optionally a language code.

If you only have a file ID then you can do a file_load on the ID to get a full file object to pass to the function.

Note that file_view_file() returns a array as expected by drupal_render() so, depending on how you are incorporating the file into your template, you may have to render the returned array using render().