I remember those film spools from my first cameras. They are very light sheet metal stock, and I think it would be hard to run them between centers, as DMIOM said, and hard to drive them as well without damaging the cheeks or the keyhole at center
My own direction would be to try to support the ends if I had to do it on the lathe by gluing one end at a time for turning to some thick aluminum sacrificail block chucked up in the four jaw. Since you are only turning the outer edge of the cheek down, you could just do one cheek (the glued one), release it from the block, glue the other cheek, and do that. Each cheek would be well supported by the backing and the center hole untouched.
What kind of glue and how much, might take some thought. working with a washer, for practice, would probably give you the info and experience.
I've used superglue for similar kinds of things. If tacked on lightly and carefully, it can be knocked loose with a sharp rap to the edge with a soft drift. Turners cement is traditional, as is shellac, I believe. Again, working out the amount and type, and method of release is important, and practice with a washer should reveal what is needed.