scripts per user on <header> for cached pages

Instead of trying to serve different assets (scripts) for different users, I'd recommend to build the scripts in a way such that they can be delivered to any user. Then use some value from Drupal.settings to decide whether or not to run the script. Authcache is capable of populating Drupal.settings selectively via Ajax/ESI. You can provide your own callbacks via hook_authcache_p13n_setting. Take a look at the Authcache User module for a good example. It is responsible for populating the contact form.

You might want to use the following skeleton:

MYMODULE.module

/**
 * Implements hook_authcache_p13n_setting().
 */
function MYMODULE_authcache_p13n_setting() {
  return array(
    'user' => array(
      'admin name' => t('Per user custom setting'),
      'admin group' => t('My module'),
      'admin description' => t('Provide custom per user data'),
      'setting' => array(
        '#setting' => 'mymodule',  // Name of the URL query parameter used when retrieving the setting value.
        '#target' => 'myModule', // Key to use when populating Drupal.settings.
        '#class' => 'MyModuleSetting', // Name of class to use when retrieving the setting value.
      ),
      'cache maxage' => 86400, // Cache-Control max-age
    ),
  );
}

/**
 * Setting handler for custom per user data of currently logged in user.
 */
class MyModuleSetting implements AuthcacheP13nSettingInterface {
  /**
   * Return custom per user data of currently logged in user.
   */
  public function get($params, $context) {
    return array(
      'foo' => 'bar';
    );
  }
}

MYMODULE.js

(function (Drupal, $) {
  "use strict";

  Drupal.behaviors.authcacheUser = {
    attach: function (context, settings) {
      if (settings.myModule) {
        // do something with settings.myModule.foo;
      }
    }
  };

}(Drupal, jQuery));

There is no session at the time when a cached page is sent to the user. So even customizing authcache_serve_page_from_cache() does not buy you much because it is impossible to tell users apart yet.

What is the use case you are trying to solve? What does require a user specific inline script in the head element?

Do you want to use User ID tracking (or any per-user dimension) or are you only interested in roles?

As a general rule I'd recommend to just set a cookie in the client during login/logout and store any ga-settings there. In order to do that just implement hook_authcache_cookie():

/**
 * Implements hook_authcache_cookie().
 */
function MYMODULE_authcache_cookie($account) {
  if ($account->uid) {
   
    $cookies['ga_uid'] = array(
      'present' => TRUE,
      'value' => $account->uid;
    );
  }
  else {
    $cookies['ga_uid']['present'] = FALSE;
  }

  return $cookies;
}

Then in your ga javascript snipped to something like this (reference):

<!-- Google Analytics -->
<script>
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');

ga('create', 'UA-XXXXX-Y', 'auto');

/* BEGIN: customization */
var userid = $.cookie("ga_uid");
if (userid) {
  ga('set', 'userId', userid);
}
/* END: customization */

ga('send', 'pageview');
</script>
<!-- End Google Analytics -->

I guess the original question is answered. Please open a new issue if there is still something unclear.