Once your Shelly finally arrives, please let us know. I know Arduino programming and configuration but I'm new to Domoticz and don't know integration with this yet. I would like to learn that myself. I see the Shelly supports MQTT and so does Domoticz. That is a simple protocol and easy to use. I ...

If you want to keep it simple, add a relay to the button on the wall. Short the lines together just like the button does. Use an ESPxxx and some kind of relay circuit.