How to add a custom field to work with my custom shipping method in woocommer

I want to calculate the delivery rate by neighborhoods without using zip code. I made a custom delivery method as below, but I can only use the default woocommerce address fields.

In the case I am using the default woocommerce city address field, it works but I want to be able to create a field named “neighborhood” and be able to use it with my custom method.

my custom method

function my_custom_shipping_init() {
    class my_custom_shipping_method extends WC_Shipping_Method {
        
        public function __construct() {
            $this->id = 'my_custom_shipping';
            $this->method_title = __('Taxa de entregar por bairros', 'my-custom-shipping');
            $this->enabled = 'yes';
            $this->title = __('Taxa de entregar por bairros', 'my-custom-shipping');
            $this->method_description = __('Plugin to calculate My Custom Shipping Cost', 'my-custom-shipping');
        }
        
        function calculate_shipping( $fields = array() ) {
            $shipping_cost = 0;
            $destination_district = $fields['billing']['bairro'];
            switch($destination_district){
                case 'Alpha': $shipping_cost = 5; break;
                case 'Beta': $shipping_cost = 7; break;
            }
            $this->add_rate(array('id'        => 'my_custom_shipping',
                                            'label'     => 'Taxa de Entrega',
                                            'cost'      => $shipping_cost,
                                            'taxes'     => '',
                                            'calc_tax'  => 'per_order'));
        }						
    } 
}
add_action( 'woocommerce_shipping_init', 'my_custom_shipping_init' );

my custom field

function my_custom_shipping_method( $methods )	{
    $methods[] = 'my_custom_shipping_method';
    return $methods;
}
add_filter( 'woocommerce_shipping_methods', 'my_custom_shipping_method' );

add_filter( 'woocommerce_checkout_fields' , 'custom_override_checkout_fields' );

// Our hooked in function - $fields is passed via the filter!
function custom_override_checkout_fields( $fields ) {
  $fields['billing']['bairro'] = array(
    'label'     => __('Bairro', 'woocommerce'),
 	'type'      => 'text',
 	'required'  => true,
 	'class'     => array('form-row-wide'),
 	'clear'     => true
  );

  return $fields;
}

And i’ve tried also

function calculate_shipping( $data = array() ) {
            $shipping_cost = 0;
            $destination_district = $data['billing']['bairro'];
            switch($destination_district){
                case 'Alpha': $shipping_cost = 5; break;
                case 'Beta': $shipping_cost = 7; break;
            }

Anyone can help?