If you're a coder I bet you could hack it to do that. It has an amazing containerized browser mode where you can browse in a remote browser via VNC, with the remote browser set up to use a WARC-writing proxy. So the general outline would be to run it locally in Docker; expose the proxy port used by the containerized browsers; and configure your own browsers to use the same proxy.

I'm not sure how much this would interfere with normal browsing -- it's not a typical usecase.