You can't 'Freeze transforms' on a blend target.
Once you make a copy of your cartoon character's head, go ahead and move it anywhere you want on the screen but don't freeze the transforms.
Usually the best thing to do, once your model is done, is to freeze the transforms on the model so that everything is zeroed out. Then make the copy or copies of the head and move them away from the model and begin remodeling them into different expressions BUT do not freeze the transforms on the blend targets (the blend shapes). If you do freeze them, you will get the same problem you are having now.
Here's something else that some people don't know; once you model the blend target and set them up so that the blend shape sliders are working, you can delete the blendshape model and it will still work.