The problem: The agents need to record a part of the call, it is not necessary to record the entire call.
The question: I would like to know if it is possible to create a button in the script that allows the agent to start and end the recording. And then that file can be downloaded.
I believe the best way to do this is to use a Web Page script component to load a custom web page and implement the buttons/controls you need in there to make requests to PATCH /api/v2/conversations/{conversationId}/participants/{participantId} to start/stop the recording. Your web app will need to implement the implicit grant (or auth code grant) to authorize the user before making API requests.