However, that post is based on the controller being in the ‘awaiting Wi-Fi’ state, i.e. broadcasting its Rachio-xxxxxx SSID. You stated that when the controller loses HomeKit connectivity, it is still connected to Wi-Fi and is otherwise functioning normally.
I don’t know whether when normally connected to Wi-Fi, it will accept the POST request and if so, whether that is sufficient to reestablish the HomeKit connection. Try it and see.