I believe your goal is as follows.

• You want to achieve the following request using Google Apps Script. Ref

    POST https://www.googleapis.com/webmasters/v3/sites/https%3A%2F%2Fwww.example.com%2F/searchAnalytics/query?key={MY_API_KEY}
    {
      "startDate": "2015-04-01",
      "endDate": "2015-05-01",
      "dimensions": ["country","device"]
    }
  

• In your situation, you want to use the following request body.

    {
      "startDate": "2021-11-01",
      "endDate": "2021-11-05",
      "dimensions": [
        "PAGE"
      ],
      "type": "DISCOVER"
    }
  

If my understanding is correct, how about the following sample script?

Sample script:

This sample script supposes that your access token can be used and you have already been able to use the API. Please be careful this.

var siteUrl = "https://www.example.com/"; // Please set the site URL.

var apiURL = `https://www.googleapis.com/webmasters/v3/sites/${encodeURIComponent(siteUrl)}/searchAnalytics/query`;
var headers = { "Authorization": "Bearer " + getService().getAccessToken() };
var payload = {
  "startDate": "2021-11-01",
  "endDate": "2021-11-05",
  "dimensions": ["PAGE"],
  "type": "DISCOVER"
};
var options = {
  "headers": headers,
  "method": "POST",
  "muteHttpExceptions": true,
  "contentType": "application/json",
  "payload": JSON.stringify(payload),
};
var response = UrlFetchApp.fetch(apiURL, options);

References:

• Search Analytics: query
• fetch(url, params)