Actually that's exactly what happened to me! :P I thought oh so cloudflare can fix this, eh. I should just reverse proxy my stuff through cloudflare.
Then I realized, oh they said cloudfront. So you could still technically keep your stuff in your S3 bucket, though you'd end up paying more for S3 originating transfer as well. In this case, cloudFRONT (the non amazon one!!) may be the better choice.
Hate that they sound the same. I even get my OWN THOUGHTS mixed up sometimes :(