Google Analytics Reports with Background Process module (non-blocking http request)
This is not possible in the manner that you described. Sure you can make a chain of asynchronous calls, the browser AJAX request to Drupal then Drupal making a non-blocking HTTP request to the GA API. However, your AJAX call won't have data to use.

The Background Process &amp; HTTPRL (check out the project page image) work by the request being one way; they are calling out to a URL that will trigger some background action that doesn't need to communicate back (e.g. some batch job or page crawler cron task). In your case, you need the GA API to communicate back with Drupal then Drupal to communicate back with the browser. There's no getting around that.