changes in atmospheric dispersion chromatic aberrations are unlikely to be the cause over such a narrow range in the red. I have certainly never seen any effects as great as this with my LHIRES. Provided you are doing a flat correction, moving the mirror should just move the spectrum up and down, and perhaps alter its brightness due to vignetting but not alter its shape. The instrument response should essentially just be the flat lamp spectrum ie for a halogen lamp in this part of the spectrum a very gentle smooth slope. (Even placing the star at a different position along the slit should not be a problem provided a flat correction is done, although it is good practise to place the star in the same position)
Is there anywhere you can upload your raw image set to (eg dropbox etc) and I could take a look ?