Sub-pages of Custom Taxonomies

I’d do it something like this.

First, register a new query var, so WP will acknowledge its existence:

add_filter( 'query_vars', function( $query_vars ) {
    $query_vars[] = 'cast';
    return $query_vars;
});

Next, add a rewrite rule, which converts the pretty permalink version of your URL behind the scenes:

add_action( 'init',  function() {
    add_rewrite_rule(
        '^movie/([^/]+)/cast/?$',
        'index.php?movie=$matches[1]&cast=1',
        'top' );
} );

(You’ll probably need to flush permalinks at this stage.)

Finally, add a clause into your archive (or whatever template you’re using) to detect the query var, and act accordingly:

if ( get_query_var('cast') ) {
    // Do what you need to do,
    // probably grab the term and use that
    // to display the term meta
}