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Ok, I took a slightly different approach and it appears to be working (filtering the episodes to only those related to the selected show), but the "Show" column is empty.

add_action( 'parse_query', 'filter_episodes_by_selected_show' );
function filter_episodes_by_selected_show($query) {
    global $pagenow, $typenow;

    if ($typenow == 'episodes' && is_admin() && $pagenow == 'edit.php' && isset($_GET['filter_show']) && $_GET['filter_show'] != '') {
        $show_id = intval($_GET['filter_show']);
        
        if (!empty($show_id)) {
            $query->set('relationship', [
                'id' => 'episodes-to-shows',
                'to' => $show_id
            ]);
        }
    }
    return $query;
}