I really love the dimensionality of the mylar. How about scanning/photographing it in it's rumpled shape, then making a high quality print of that? The print would be flat, but it would look like the piece itself. That might be better than flattening the original. It would only work if it captures the light and sheen of the mylar, though, and such images can't always do it. Worth checking out, though. As Jim said, it's possible this could gain value as a historical artifact. Perhaps storing the original is a good idea. I'm having fun wondering whether these would look better on a white background, black, or something like brushed steel. Another thing you might try is contacting a space museum (she writes....as if there's one on every block), and talking with their curator*.
* I really like the idea that someone out there might have this job. Bob: 'I'm a plumber, how 'bout you?' Fred: 'Space museum curator'.