Unique design inside a specific category’s url

There is a filter called theme_page_templates which you can use to force the use of a certain template under certain circumstances. Like this:

add_filter ('theme_page_templates', 'wpse263827_filter_theme_page_templates', 20, 3 );
wpse263827_filter_theme_page_templates ($page_templates, $this, $post) {
  if (condition based on $post) $page_templates="path-to-template.php";
  return $page_templates;
  }

So, you can place the template file wherever you want and assign it when the page is in that certain category. No need to touch the url structure.