We're a small team working in stealth on this exact challenge. Shoot me a note if you're interested in hearing more or getting involved. itshelikos@gmail.com
As soon as Facebook realizes you're a risk, you'll get a C&D ordering you to stop accessing their servers. These typically have the force of law under the CFAA.
You won't access their servers, but just read the page that the user already downloaded? You'll still get nailed under the Copyright Act.
"User agents" in the sense used by the OP are as old as the internet itself. There's an active, serious, and quiet effort to abuse outdated legislation to ensure that they never become a problem.