Signing the raw image data wouldn't require also signing the EXIF metadata. For privacy, you could later strip out everything except the raw image and the camera's signature and still prove that the image is unaltered.
Even if you didn't know who owned the camera, you could identify other pictures taken by that same camera, and information in those photos might let you figure out who owns the camera.
If you sign two images with that private key, people will know for certain that you created both of them. This is the whole point of signing something. No one else can fake it because they don't have the private key, but EVERYONE can verify it because they DO have the public key. In fact, they don't even need the public key because they can extract it from the signature and the signed image.
If authenticity later becomes an issue you can produce the original.