Sorry, I forgot to post the answer when I found it. In actual fact there was nothing wrong with the authentication configuration, it was to do with my referencing Context.User too early in Global.asax.
In the new integrated mode the authentication works differently to classic mode and in the Application_AuthenticateRequest event in Global.asax the authentication hasn't happened yet so Context.User is null hence the exception I got and the fact that the log shows IsAuthenticated as false. I moved the line of code that calls Context.User into Application_PostAuthenticateRequest and it's working now.