I am making a custom shipping extension, which uses the prices of shipping methods from another site

Currently the shipping methods add to the shipment combo box on the checkout page
When any of the methods get selected, the shipping details do not add to the sub total price / total price of the checkout (the attachment shows this)


Here is my current code for adding the shipping methods to the combo box.

Quote
function getQuote($address) {
   
        $quote_data = array('custom_method' => array(
                'id' => 'custom_shipping.custom_method',
                'title' => 'Custom Method',
                'cost' => 5,
                'tax_class_id' => 0,
                'text' => $this->currency->format('5')));
       
   $method_data = array(
      'id'         => 'custom_shipping',
      'title'      => 'custom shipping',
      'quote'      => $quote_data,
      'sort_order' => 0,
      'error'      => false);
      
   return $method_data;
}