asked to update wordpress database at admin login

I am being asked to update the wordpress database at the wp-admin login page. It seems despite my best efforts, the site WP version has been updated to the next major version and cannot be reverted (a byproduct of the siteground migration to their new site tools platform). I am concerned that if I update the WP database the site will crash as the template is old and I am in the middle of negotiating what next steps to take with the client. How do I work around this problem?

The page I need help with: [log in to see the link]