If it exists the Ring doorbell button won’t be a switch (output) it will be a binary sensor (input).
The entities associated with your Ring integration can be found on the page Configuration / Integrations / Ring.
If such a binary sensor exists you will then need this python script to alter the state of your Ring binary sensor to simulate a button push (binary sensors are not normally meant to be changed by Home Assistant, but ‘sensed’ by the integration instead):
The other option is to write an automation triggered by the Shelly that does the same actions as your Ring doorbell (e.g. camera snapshot and notification).