Wordpress - Using save_post to replace the post's title

This simplest method would be to edit the data at the point it's inserted, rather than updating it afterwards, using wp_insert_post_data instead of save_post. This works on creating a new post or updating an existing post without change. It also avoids the danger of creating an infinite loop by triggering update_post within save_post.

add_filter( 'wp_insert_post_data' , 'modify_post_title' , '99', 1 ); // Grabs the inserted post data so you can modify it.

function modify_post_title( $data )
{
  if($data['post_type'] == 'rating' && isset($_POST['rating_date'])) { // If the actual field name of the rating date is different, you'll have to update this.
    $date = date('l, d.m.Y', strtotime($_POST['rating_date']));
    $title = 'TV ratings for ' . $date;
    $data['post_title'] =  $title ; //Updates the post title to your new title.
  }
  return $data; // Returns the modified data.
}

I had the exact same need, so I wrote this function - which works. Modify it to your needs. Hope this helps.

// set daily rating title
function set_rating_title ($post_id) {
    if ( $post_id == null || empty($_POST) )
        return;

    if ( !isset( $_POST['post_type'] ) || $_POST['post_type']!='rating' )  
        return; 

    if ( wp_is_post_revision( $post_id ) )
        $post_id = wp_is_post_revision( $post_id );

    global $post;  
    if ( empty( $post ) )
        $post = get_post($post_id);

    if ($_POST['rating_date']!='') {
        global $wpdb;
        $date = date('l, d.m.Y', strtotime($_POST['rating_date']));
        $title = 'TV ratings for ' . $date;
        $where = array( 'ID' => $post_id );
        $wpdb->update( $wpdb->posts, array( 'post_title' => $title ), $where );
    }
}
add_action('save_post', 'set_rating_title', 12 );

Here's a solution that uses a static variable to prevent an infinite loop. This allows you to safely call wp_update_post() inside of a function that is hooked to save_post.

function km_set_title_on_save( $post_id ) {

    // Set this variable to false initially.
    static $updated = false;

    // If title has already been set once, bail.
    if ( $updated ) {
        return;
    }

    // Since we're updating this post's title, set this
    // variable to true to ensure it doesn't happen again.
    $updated = true;

    $date           = get_post_meta( $post_id, 'rating_date', true );
    $date_formatted = date( 'l, d.m.Y', strtotime( $date ) );

    // Update the post's title.
    wp_update_post( [
        'ID'         => $post_id,
        'post_title' => 'TV ratings for ' . $date_formatted,
    ] );
}
add_action( 'save_post', 'km_set_title_on_save' );

Note: To limit this functionality to a certain post type, use the save_post_{$post->post_type} hook instead of save_post.