The process would be that the browser send a request to MITMproxy and then generate a request that gets sent to target proxy server which isn't controlled by us. The proxy server would send a response to MITMproxy which would then relay that response to the browser.
How would I go about doing this?
You can do this with the -F flag to mitmproxy, which forwards proxy requests to an upstream server. Take a look at the docs here: