Hmm, right you are - Links running on the same system has no issues. I suppose that, unfortunately, we probably have to chalk this one up to ELinks being no longer actively maintained... :/ Still, this'll do in a pinch - both my public-access *nix shell accounts have Links as an option, so I can always post from there.
Edit: Even more peculiarly, when I copy the existing cookie over from Pale Moon to ELinks, it works perfectly. It just doesn't seem to want to accept the new one handed out by the login form. How odd.