In your situation, if <?!=include('Stylesheet'); ?> is included in index.html file, how about the following modification?

From:

function doGet() {
  return HtmlService.createHtmlOutputFromFile('index');
}

function include(filename) {
  return HtmlService.createHtmlOutputFromFile(filename).evaluate().getContent();
}

To:

function doGet() {
  return HtmlService.createTemplateFromFile('index').evaluate();
}

function include(filename) {
  return HtmlService.createHtmlOutputFromFile(filename).getContent();
}

Note:

• If filename of include(filename) has the scriptlets, please modify as follows.

    function include(filename) {
      return HtmlService.createTemplateFromFile(filename).evaluate().getContent();
    }
  

• In your script, it seems that Web Apps is used. In this case, when you modified the Google Apps Script, please modify the deployment as a new version. By this, the modified script is reflected in Web Apps. Please be careful this.

• You can see the detail of this in the report of "Redeploying Web Apps without Changing URL of Web Apps for new IDE".

References:

• createTemplateFromFile(filename)
• createHtmlOutputFromFile(filename)