Writing a reverse proxy/loadbalancer from the ground up in C, pause to regroup: fixed it!
It took a bit of work, but the bug is fixed: rsp now handles correctly the case when it can't write as much as it wants to the client side. I think this is enough for it to properly work as a front-end for this website, so it's installed and running here. If you're reading this (and I've not had to switch it off in the meantime) then the pages you're reading were served over rsp. Which is very pleasing :-)
The code needs a bit of refactoring before I can present it, and the same bug still exists on the communicating-to-backends side (which is one of the reasons it needs refactoring -- this is something I should have been able to fix in one place only) so I'll do that over the coming days, and then do another post.