Oh, sorry for taking a while to get back to this one.
A hair shader is about combining together approximations for all the different ways light could pass through a forest of glass tubes. Some of those are within one hair, like "reflecting off (R)", "refracting through one side and out the other (TT)", or "refracting in, bouncing around 3 times, then refracting out (TRRRT)". Some of them are about bouncing back and forth between other nearby hairs. Of all these approximations, the only thing that would really make sense to affect with the tube normal is the very first "(R)" lobe. So in theory, you'd want to turn off just that lobe and replace it with a simple rough reflection ... however I don't think there's really any way to do exactly this with the Arnold ( it would be sort of like setting `spec` to 0 and just keeping `spec2` on the Arnold shader, but I think that would have other side effects that would break the rest of the shading ). Assuming you don't have your own hair shader to hack with, I guess you could just turn down a bit all the specular on the Arnold shader, and add an extra spec lobe on top with the tube normal ... it won't be particularly accurate, but maybe it would get you close to your desired look?
The other option would be just use actual tubes, with a glass shader on them. Hair shaders are just an approximation to that ... the only reason we use hair shaders is because rendering millions of tiny glass tubes with many inter-reflections is insanely expensive and noisy.
Anyway, I'm still not sure what the goal here is ... I'm still not picturing a situation where the normal variation across the width of a hair would be visible, unless the hair is thick and cartoony, or we're looking through a microscope.
-Daniel