Got it. For some reason, it wasn’t wrapping the message in the correct class. So I changed it:
function custom_login_message() {
$message = "<p class='message'>Welcome, if you haven't already, you need to <a href='LINK'>request</a> to be accepted to alpha viewing.</p><br />";
return $message;
}
add_filter('login_message', 'custom_login_message');
?>