Whilst you’ve got the right idea, it won’t work in this case. Taking a look at the source code, the class is simply instantiated without an assign. There’s no way to get a hold of $this from outside the class, and so no way for you to remove the hooks it attaches itself to.
This is why I’m a strong advocate of static methods in plugins. I would suggest contacting the author or TinyMCE Templates & asking if he would consider moving to this style of coding, or at least assigning the class instance to a global variable so that your code will indeed work.
Short term fix? Edit the plugin file & remove the add_action line, or use remove_all_actions( 'post_submitbox_start' ), though do so with extreme caution!