Let's start over. If you bought the game on disc, you shouldn't need to reinstall the base game.
When you put the disc in for the first time, it will install the episodes. I think it does it in order, so that if you start with the Prologue and Paris, it will have ideally installed the later episodes by the time that you 'get' to them and you don't really notice that it's installing.
Make sure to completely close the game and then launch it again. If Colorado still says 'get access', please check the below.
Go to 'My games and Apps' from the Xbox Dash and find HITMAN under 'Games'. Hit the menu button on your controller and then choose 'Manage game'. That is where it will show you what is installed. You have the base game at the top and then all of the episodes underneath. Are they all listed there and showing installed? Is there anything listed under 'Ready to Install'?