Run jQuery after WooCommerce AJAX shipping method update
Correct Custom Event: updated_checkout
The code snippet in question was listening to the incorrect Woocommerce custom event.
The correct custom event to listen to in this case is updated_checkout.
When selecting a new shipping method, Woocommerce fires an ajax call to update the cart totals to reflect the price of the shipping method selected, then triggers updated_checkout.
So if updated_checkout is listened to, rather than updated_shipping_method, the script is fired as intended.