Drupal caching prevents cookie read
I think you have have a hard time doing this with caching, since you want to create two versions of the same page.

I haven't tried this, but it should be possible to let your style.css not be an actual css file but a drupal callback that returns the styles based on the cookie. You would need to include the "stylesheet" as you would do styles for IE. You wouldn't be able to cache this, but if you really need to performance, you could do it in a php file and skip the drupal bootstrap.

An alternative option would be to let the js itself execute the styles, based on the cookie.

None of these options are ideal, but they should work.