Partial Requests To The Rescue

While we don't want to get all the data, a GET request does contain the Content-Length header we're after. Fortunately we don't have to get all the data, we can make a partial request. Using the Content-Range header we can instruct GitHub to send us only a small amount of the data but all the headers.

Show Me Some Code!

curl -L -i -r 0-1 https://github.com/release/download/url

The important part is -r 0-1. This causes cURL to send a Content-Range header with the value 0-1 which means "get me the first byte only". Now we have easy access to the Content-Length header and we only have to download an additional byte.

Isn't There An API For This?

Yes. GitHub does expose an API that allows you to fetch a JSON object that represents everything you could want to know about repo downloads but I wanted something simple and easy to access and that didn't require decoding a JSON object.