I can't find any reference to an API that enables Rest API clients to export an existing Google Sheet to a csv file.
https://developers.google.com/sheets/
I believe there should be a way to export them.
The following URL gives you the CSV of a Google spreadsheet per sheet. The sheet must be accessible by the public, by anyone with the link (unlisted).
The parameters you need to provide are:
https://docs.google.com/spreadsheets/d/{{ID}}/edit
)https://docs.google.com/spreadsheets/d/{{ID}}/gviz/tq?tqx=out:csv&sheet={{sheet_name}}
With that URL you can run a GET-request to fetch the CSV. Or paste it in your browser address bar.