Didn’t look at the .blend.
If you just want to do object level transforms on the object, you can remap the normal into object space:
If you want to rig it or use modifiers, you have to bake the texture to a UV texture.
Your nodes aren’t quite right, by the way. “Green”, as the Y component of your normal, actually goes from -1 to 1, not 0 to 1. A math/abs node plugged in there should do the trick, otherwise you’re mixing the side view into the back.