use below line in app/Design/frontend/Your/Theme/Magento_Theme/layout/default.xml
<move element="header.links" destination="footer_links" after="-"/>
If you change any of xml file and your cache is enabled then you must have to run below command
php bin/magento cache:clean
If you can overwrite this 3rd party module (and most probably you can), you could post a message from iframe to parent frame using Window.postMessage() after logging the user in or out (see: https://developer.mozilla.org/en-US/docs/Web/API/Window/postMessage). Then you could write a JS script somewhere on your page to listen to such message, and either ...