The HPC usage scenario is exactly the reason why I played with UML a few years ago. Many HPC clusters nowadays completely lock down the network and only allow in/outbound SSH connections to whitelisted IP networks. On the other hand, users are only given an unpriviledged account. Typically, users set up proxies on a per-application level (i.e. for git, svn, to sync their codes with the outer world). I found this very frustrating and wanted to use any usual Linux command line tools. My idea was a UML based virtual machine (ie. running a Linux distribution within UML) and have it connected to the outer world by tunneling a single slirp and VDE connection. VDE is quite amazing because it marries unix pipes with ethernet-level networking. It even allows to run ethernet switches as processes and all that. My codes based on https://github.com/jpetazzo/sekexe
Turned out: It basically worked, but it was a bit painful to setup and to manage.
I'd have thought it would be relatively unusual not to be able to use HTTP into a cluster login node, but I'm puzzled why you can't "sync code" with ssh otherwise. If you're prevented from doing things you need to work, it sounds like an institutional politics problem; subverting controls is likely to cause more friction as well as presumably wasting resources due to inefficiency.
Turned out: It basically worked, but it was a bit painful to setup and to manage.