unable to execute Apps Script API Executable from another Apps Script as another user

Question

I created Apps script and deployed it as an API Executable with the follow settings:

Web App
Execute as: ME
Who has Access: Anyone within the domain

API Executable
Who has Access: Anyone within the domain

Once deployed, I get 2 URLs, one for the Web App and one for API Executable. The test function in that API Executable is:

function saveObservation(bookId, name)
{
  console.log (bookId + ":" + name);
  //var book = SpreadsheetApp.openById(bookId).getSheetByName(name);
}

In another Apps Script I have a code which invokes the API Executable URL

  var token = ScriptApp.getOAuthToken();
  var options = {
    "method": "POST",
    "headers": { "Authorization": "Bearer " + token },
    "contentType": "application/json",
    "payload": JSON.stringify({
      "function": "saveObservation",
      "parameters": [bookId,sheetName],
      "devMode": "true"
    }),
    "muteHttpExceptions": true
  }

  var rest = UrlFetchApp.fetch("https://script.googleapis.com/v1/scripts/ApiExecuteID:run", options)
  console.log("rest "  + rest)

If I execute this code as myself, everything works fine. When I then share the spreadsheet and some one else executes the code, the script fails and in the logs I see

rest {
  "error": {
    "code": 403,
    "message": "The caller does not have permission",
    "status": "PERMISSION_DENIED"
  }
}

  

If I invoke the Web app URL as opposed to the API Executable URL, it works fine.
The code is not that clean though since I have to maintain the doPost() function to then call the desired function.

So what am I missing in order so that the API Executable can be called by domain user?