I set shipping total and total via AJAX:

add_action('wp_ajax_wc_shipper_set_rate', 'wc_shipper_set_rate');
add_action('wp_ajax_nopriv_wc_shipper_set_rate', 'wc_shipper_set_rate');

function wc_shipper_set_rate(){
    $object_id = strval($_POST['object_id']);
    $order_id = intval($_POST['order_id']);
    $rate = false;

    //Find rate by object_id
    $rates = WC()->session->get('wc_shipper_rates');
    foreach ($rates as $_rate) {
        if ($_rate['object_id'] == $object_id) {
            WC()->session->set('wc_shipper_chosen_rate', $_rate);
            $rate = $_rate;
        }
    }

    if($rate){
        //Calculate new totals
        $totals = WC()->cart->get_totals();
        $new_total = floatval($totals['total']) - floatval($totals['shipping_total']) + floatval($rate['amount']);
        
        // Set cart totals
        WC()->cart->set_shipping_total($rate['amount']);
        WC()->cart->set_total($new_total);
        
        $totals = WC()->cart->get_totals();
        WC()->cart->calculate_shipping();
        WC()->cart->calculate_totals();
        
        // Change shipping total in order
        
        $order = wc_get_order($order_id);
        $order->set_shipping_total($rate['amount']);
        $order->set_total($new_total);
        $order->save();
        
        // Change shipping total in session
        WC()->session->set('shipping_total', $rate['amount']);
        WC()->session->set('total', $new_total);
        WC()->session->set('cart_totals', $totals);

        update_post_meta($order_id, '_wc_shipper_object_id', $rate['object_id']);
    }

    echo json_encode(['status' => 'ok', 'totals' => $totals], JSON_UNESCAPED_UNICODE);
    exit;
}

In WP-ADMIN new totals and the shipping total are visible:
New totals in the order

But on the checkout page there are old totals. New totals are visible after double page refresh.
Old totals

The WC method is useless.

WC()->cart->calculate_shipping();
WC()->cart->calculate_totals();

I use the last release WooCommerce and WordPress.