ASP.NET MVC with Forms Auth and WebApi with Basic Auth

I have a WebApi using Basic Auth nicely. And I have an MVC site using Forms Auth nicely. But here's the catch:

Client X has a dedicated database with any number of Contacts and Products. The MVC site is a dedicated site for them (via {clientId} routing), which allows their Contacts to log in (via Forms Auth) and place orders for their products. The Contact must be Form-ly logged in to place an order.

The product orders (need to) hit the WebApi to be recorded in the Client's database.

But since the WebApi uses Basic Auth to validate the Client, not the Contacts who placed the orders, every request comes back is 401 - Unauthorized.

I've checked out ThinkTecture as suggested by a number of posts here on SO, however it doesn't get me what I need because I'm not looking to allow Forms Auth in the WebApi. I don't want to authenticate the Contact from the Client's database in the WebApi, I want to authenticate the Client in the WebApi.

Has anyone come across a similar scenario and am I missing something glaringly obvious? Perhaps I need to implement both Forms and Basic on the site?

The very standard Api call I'm making from the site (where the UserName and Password are the Client's, not the Contact's):

var clientId = new Guid(RouteData.Values["clientId"].ToString());
var baseUrl = ConfigurationManager.AppSettings["ApiBaseAddress"];
var authHeader = Convert.ToBase64String(Encoding.ASCII.GetBytes(String.Format("{0}:{1}", _shoppingCartSettings.UserName, _shoppingCartSettings.Password)));
var requestUrl = String.Format("api/{0}/inventory", clientId.ToString());

var httpWebRequest = WebRequest.Create(baseUrl + requestUrl);

httpWebRequest.Headers.Add(HttpRequestHeader.Authorization, "Basic " + authHeader);
httpWebRequest.Method = "GET";
httpWebRequest.Accept = "application/json";
httpWebRequest.ContentType = "application/json";

try
{
    using (var httpWebResponse = httpWebRequest.GetResponse())
    {
        // we never get here because of a 401
    }
}
catch (WebException ex)
{
    using (var httpWebResponse = ex.Response)
    {
        // we always get here
    }
}

If I set up a separate test client and make the same call, it works great :/