How to handle authorization for Apps Script from Chrome extension - google-apps-script

Question

I try to execute a Google Apps Script from my Chrome extension. My Google Apps Script uses Spreadsheet Service - it reads data from the user's spreadsheet, url of which is provided by user to my extension and then sent by extension to GAS. So, I've deployed Google Apps Script as web app, which is executed by a user accessing the web app and can be accessed by anyone. The problem is that when the script is run for the first time by a new user, the user can't see the authorization request so the script can't access the spreadsheet and send data to the extension. Everything works fine only if a user runs the script manually in the browser - only then the authorization request appears.
My question is how extension can trigger the authorization request to appear automatically?
I have read this question with answers Running Apps Script from Chrome extension requires authorization but there are only instructions on how to avoid authorization, which in my case can't be avoided. I would appreciate any help with this.
Update: I'm trying to use getAuthorizationUrl() as advised in the comments and now I'm stuck again. Here is the code from the Google Apps Script:
function doGet(e) {
var url=e.parameter.url;
var ss = SpreadsheetApp.openByUrl(url);
var authInfo = ScriptApp.getAuthorizationInfo(ScriptApp.AuthMode.FULL);
var status=authInfo.getAuthorizationStatus();
if (status=="NOT_REQUIRED") {
var sheet = ss.getSheets()[0];
var data = getData(sheet);
if(!data) {
data = '';
}
return ContentService.createTextOutput(JSON.stringify({'result': data})).setMimeType(ContentService.MimeType.JSON);
} else {
var authUrl=authInfo.getAuthorizationUrl();
return authUrl;
}
}
In both cases responseType of my XMLHttpRequest will be "", so how can my extension distinguish between 2 possible responses? I need something like this in the content script (don't know what to put inside if):
xhr.onreadystatechange = function(response) {
if (xhr.readyState === 4 && xhr.status == 200) {
try {
if (??response is authUrl) {
//do redirect;
}
else {
//process the data from the spreadsheet
}