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
}