It is NOT possible to retrieve the refresh token of an App Script application. Apps Script projects are automatically generated and internally managed by Google, and as far as OAuth goes, they only expose access tokens via the ScriptApp built-in service.
If you plan on making your Add-on publicly available, then be aware that passing an end-user's access token that uses restricted scopes (Gmail, Drive, etc.) to a 3rd-party server will likely trigger a security assessment which will run you anywhere from US$10,000.00 to US$75,000.00 annually.
See Google's support docs for more information:
https://support.google.com/cloud/answer/9110914?hl=en#zippy=%2Csecurity-assessment