HttpContextWrapper and HttpContextBase, as explained here, were introduced to make HttpContext more mockable/testable.
I\'m trying to use it with S#arp Architecture,
My MVC Controllers are set up to accept an HttpContextBase argument in the constructor
You gotta be doing something extremely wrong here, so stop before it's too late and damage has been caused (material, moral and human casualties :-)). You already have the HttpContext inside the controller.
Don't register any HttpContexts in your DI framework. The HttpContext handling is the job of ASP.NET.