Append Custom Shipping method to list of methods

In Magento 2.3 I want to append a shipping method to my collection of methods but it’s not working. This is what I’m doing:

1. I’m selecting the cheapest rate from the collection of rates with
   
$cheapestRate = $this->getResult()->getCheapestRate().
2. Then, I reset the original collection with $this->getResult()->reset()

3. I append the cheapest rate to the now clean collection of methods with
   
$method->setCarrier($cheapestRate['carrier']);
$method->setCarrierTitle($cheapestRate['carrier_title']);
$method->setMethod($cheapestRate['method']);
$method->setMethodTitle($cheapestRate['method_title']);
$method->setCost($cheapestRate['cost']);
$method->setPrice($cheapestRate['price']);
$result->append($method);
$this->getResult()->append($result);

4. I’m displaying the cheapest rate so far. Now, I want to add a custom method with a cost of $0.00. This is what I do:
   
private function addCustomShippingToMethods($method, $result) {
$method->setCarrier('custom_shipping');
$method->setCarrierTitle('custom_shipping');
$method->setMethod('custom_shipping');
$method->setMethodTitle('custom_shipping');
$method->setPrice(0);
$method->setCost(0);
$result->append($method);
return $this;
}

5. So, to insert the custom method in step 4, I call addCustomShippingToMethods right after the last line in step 3 like this
   
$this->addCustomShippingToMethods($method, $result);

The problem is that I only see the custom method in the checkout page but not the cheapest one. If I remove the call to add the custom method then I only see the cheapest one.

In a few words, all the shipping options I want to see in the checkout page are the cheapest rate and the custom shipping at $0.00.

What am I doing wrong?

Thanks for any advice.