Sorry if this comes off as confusing.
I have written a script using the NodeJS request module that runs and performs a function on a website then returns with the data.
Also, if you have control over the proxy server, can you log the requests being made to the final server?
Try writing your own proxy (a reverse one) and host it somewhere. Instead of requesting to https://target.com, to a request to your http[s]://proxy.com/ and let the reverse proxy do the work. Also, remember to disable X headers on the implementation as it will change the request headers
Reference for node.js implementation:
https://github.com/nodejitsu/node-http-proxy
Note: let me know about the questions I made in the comments