Is it possible to apply Authcache caching to elements in page.tpl.php?
I recommend to redo the page with blocks (or panels). The Authcache User Cookie example is rather basic and does not demonstrate how to deliver fragments from the custom front controller.

Regrettably there is still not enough documentation on how to roll your own custom fragment renderer. Look at the api.php files and read through the numerous submodules to get the picture.

Last but not least, there are some (closed) support requests in the ticket tracker covering this issue, e.g. #2780485.