Ticket Data to update survey data

The only thing I can think of is have API pull ticket data (which should include the response ID), and then it push the necessary update back to survey data based on that response ID via another API call. I think this API work would need to happen outside of Qualtrics, but maybe someone will have a better idea.

Thank you for the suggestion, I'm reading up on the API to see if I can put something together.

Radam
As mentioned by InessaG the only possible way is through API. I would like to provide more specifics here.
You can create a ticket event based workflow. Within which you can choose status changed. Have the condition to say if status is Approved. Here, kindly include an embedded data in survey flow called ResponseID to record the response id and then include it in the ticket data as well while you create ticket task. Post which use this in the API task to update the response.
Once done include an webservice task: API task of update response. Configure to update an embedded data to approved. If the above conditions are satisfied. Your URL would look like: https://yul1.qualtrics.com/API/v3/responses/${tck://Field/responseId}.
This is how your configuration would look:
Status : Approved in your case.
Hope this helps!