You could plausibly use cloudfront, I don't know what tumblr's policies are. But these aren't technical limitations! I don't know why Tumblr retricts you from

You could plausibly use cloudfront, I don't know what tumblr's policies are. But these aren't technical limitations! I don't know why Tumblr retricts you from using cloudflare, presumably because they want to control your content. My two cents, don't use a free hosting service. S3+cloudfront is dirt cheap and will do the trick.