Setting Shipping Tax programatically

Hi,

We’re trying to change the Tax class of an order based on if a field is filled on the checkout page. We have successfully made this happen for the products to set the tax class to “reduced-rate” but we’re struggling to also change the shipping tax class based. Any hints?

Here’s the code for reference:

add_action( 'woocommerce_before_calculate_totals', 'change_tax_class_based_on_custom_input' );
function change_tax_class_based_on_custom_input(  $cart ) {
    $billing_ic = (float) WC()->session->get('billing_ic');
	if ( $billing_ic ) {
		foreach( $cart->get_cart() as $cart_item ){
        $cart_item['data']->set_tax_class("reduced-rate");
		}
    } 
}