How can I invoke default model binding within a custom binder in MVC4 Web API?

I recently posed a question here on Stack Overflow about finding the right extensibility point for model binding in the Web API beta. Using a custom provider / IModelBinder gives me full control of model binding and access to the value providers.

However, it's really too much control. I just want to control how a couple of values are mapped, and don't want to hand-code model binding that would've otherwise worked just fine.

Ultimately, this is what I'd like to be able to do:

public class MyCustomModelBinder : IModelBinder
{
    public bool BindModel(HttpActionContext actionContext, ModelBindingContext bindingContext)
    {
        // Invoke default model binding

        // Pull out custom values
        var value1 = bindingContext.ValueProvider.GetValue("value1");
        var value2 = bindingContext.ValueProvider.GetValue("value2");

        bindingContext.Model.Value1 = DoCustomStuff(value1);
        bindingContext.Model.Value2 = DoCustomStuff(value2);

        return true;
    }

    ... // Define DoCustomStuff
}

It looks like it's pretty hard to get ahold of a default provider for the WebAPI. Does anyone know if it's accessible and how it should be accessed here in a custom model binder?