I think that weird clamp-thing Rey is holding is a regular, lighted blade Graflex, with a thin foam covering the Graflex for comfort. It looks like it goes from above the grips, over the clamp, and up to where the glass eye and button are.
Yup, must be the "cloth" covering they talked about. Strange how noticeable it is in the pics here yet I never noticed it in the theater watching the film. Probably something that will now bug me in subsequent viewings since I know where to look for it.