Adding site title to overlay menu

To accomplish this I would start by adding a second Site Title to the navigation menu.
Select the original Site Title block and in the block options click ‘Duplicate’. You will now have two Site Title’s. Drag the second Site Title to the Navigation block.

Now you will need some custom CSS to hide the Site Title in the menu on larger screens:

@media (min-width: 600px) {
  nav .wp-block-site-title {
    display: none;
  }
}

As the classic customizer/additional CSS link is not available for block themes, you can go directly by entering the URL https://www.yourdomain.com/wp-admin/customize.php in your browser address bar, and add the custom CSS under ‘Additional CSS’ and save/publish.