That would not be easy because the buttons are added to the page from a different file.
The login required text is from here: discourse/app/views/static/login.html.erb at main · discourse/discourse · GitHub. Adding a div there should not be a problem.
The button (and surrounding div) come from here: https://github.com/discourse/discourse/blob/master/app/assets/javascripts/discourse/templates/static.hbs. That template is also used for pages like the FAQ and TOS, so we cannot change the surrounding div’s class, or add anything else (like a new div) that is specific to the login required page.