The search results sorting filter contains four sorting criteria options: title (post_title), price (price_from), rating (rating), available date (av_date_from).
However, the plugin also supports sorting by several more parameters: post publication date (post_date), post modified date (post_modified), menu order (menu_order).
Below is an example of adding a new sorting parameter to the filter (requires BA Book Everything 1.8.1 or higher).

add_filter('babe_search_filter_sort_by_args', 'child_theme_babe_search_filter_sort_by_args', 10, 1);
function child_theme_babe_search_filter_sort_by_args( array $filters ): array
{
    return array_merge( [
        'post_date_desc' => __( 'Sort by...', 'your-child-theme-textdomain' ),
        'post_date_asc' => __( 'Sort by...', 'your-child-theme-textdomain' ),
    ], $filters );
}

add_filter('babe_search_filter_to_get_posts_args', 'child_theme_babe_search_filter_to_get_posts_args', 10, 1);
function child_theme_babe_search_filter_to_get_posts_args( array $args ): array
{
    if( $args['search_results_sort_by'] === 'post_date_desc' ){
        $args['sort'] = 'post_date';
        $args['sort_by'] = 'DESC';
    }
    if( $args['search_results_sort_by'] === 'post_date_asc' ){
        $args['sort'] = 'post_date';
        $args['sort_by'] = 'ASC';
    }
    return $args;
}

add_filter('babe_search_result_args', 'child_theme_babe_search_result_args', 10, 1);
function child_theme_babe_search_result_args( array $args ): array
{
    if( empty($_GET['search_results_sort_by']) ){
        $args['search_results_sort_by'] = 'post_date_desc';
    }
    return $args;
}