Proxy HTTP requests while modify their bodies

Image this scenario:

1. You have a web server that serves static frontend files. For example, www.google.com.
2. You can not access and modify those files (can't change CSS, can't change HTML, etc.)
3. However, you want to re-style the web page using CSS. For example, restyle how www.google.com looks like.
4. You would set up a proxy server to pass requests and responses. Responses would be read and modified by the proxy server before passing back. For example, adding custom CSS.

The whole diagram would look something like this:

And before someone accuses me of trying to hack someone, I am simply trying to go around a limitation of AWS Cognito built-in login-page limitations. For example, you can not change the background of the web page.