Magento one click checkout

i try to implement a one-click checkout for Magento-1.x, but if i use my Checkout, they have after a buy the problem that the shipping method is not selected.

Here is the Code from my Checkout Controller.

    <?php


    class MCP_Debitapi_CheckoutController extends Mage_Core_Controller_Front_Action
    {
        public function oneClickAction()
        {

            /**
             * @var Mage_Catalog_Model_Product $product
             */
            $options = [];

            $product = Mage::getModel('catalog/product')->load((int)$_POST['product']);
            /**
             * create a cart to work with
             * @var Mage_Checkout_Model_Cart $cart
             */
            $cart = Mage::getModel('checkout/cart');
            $cart->addProduct($product,$options);
            $cart->save();
            /**
             * @var Mage_Checkout_Model_Type_Onepage $checkout
             */
            $checkout = Mage::getSingleton('checkout/type_onepage');
            $checkout->getQuote()->clearInstance();
            $checkout->initCheckout();
            /**
             * @var Mage_Customer_Helper_Data $customerHelper
             */
            $customerHelper = Mage::helper('customer');
            $billingAddress = $customerHelper->getCustomer()->getDefaultBillingAddress();
            $shippingAddress = $customerHelper->getCustomer()->getDefaultShippingAddress();
            if(!$billingAddress) {
                Mage::throwException('cannot create one-click order without default billing address');
                return false;
            }

            $checkout->getQuote()->getBillingAddress()->importCustomerAddress($billingAddress)->setSaveInAddressBook(false);
            $checkout->getQuote()->getShippingAddress()->importCustomerAddress($shippingAddress)->setSaveInAddressBook(false);
            $checkout->getQuote()->getShippingAddress()->setShippingDescription('Fixed')->save();
            $checkout->getQuote()->getShippingAddress()->setCollectShippingRates(true);
            $checkout->saveShippingMethod('flatrate_flatrate');
            $checkout->savePayment(array('method'=>'mcpdebitapi'));
            $checkout->getQuote()->collectTotals();
            $checkout->saveOrder();
            $cart->truncate();
            $cart->save();
            $cart->getItems()->clear()->save();
            $checkout->getQuote()->setIsActive(false)->save();
            $this->_redirect('checkout/onepage/success',array('_secure'=>true));
        }

    }

How i can find out and set the correct shipping method?