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.
I understand that's not technically "your problem", but since you mentioned it, just thought I'd let you know :)
It's unfortunate, because tumblr really is a great blog hosting software and service, if you don't use the social aspects of it.