I used @paulsm4 suggestion and found this page on Microsoft's site. Using that, I ended-up with this, and it's working great:
ChannelFactory<IMyService> myFactory = new ChannelFactory<IMyService>(new WebHttpBinding(), myRemoteAddress);
myFactory.Endpoint.Behaviors.Add(new WebHttpBehavior());
myClient = cf.CreateChannel();
I realize this doesn't exactly solve the original question, but I don't think there's a way to read ALL of the binding info from an alternate config file. If there is, I welcome someone to answer the original question.