Setting a Twig to a specific Form Container
You could try to add your own template suggestion via hook_theme_suggestions_HOOK_alter, which would mean hook_theme_suggestions_container_alter. There you might target the container by id or some other attribute (id does not seem promising here).

The container.html.twig does only provide a wrapping div around other content. While you could add more html to this, you might have to think about if container.html.twig is the right place to start here.