I've never actually completely implemented a Squid server before, but I came close near April 1st when I tried to implement the famous Upside Down Ternet for April Fools (stopped because I got really sick and ran out of time).
If it were me, I would look into doing something very similar: install Squid onto the server, route your internet through that server (or at least for particular URLs) and then use a script to download the file(s) onto the server automatically when you request them from your computer, and push them through the server and down to you (the client).
That's a bunch of theory that I unfortunately can't backup with lots of experience. But good luck!
Here's a couple resources on Squid to get you started: