OP, as you may already be aware, Jobcentres now apparently have targets (moronic workplaces love them). In other words, they have a sanction quota. So they take any chance they get. Sadly, you're completely right, you should've lied. However, there's every chance the Jobcentre will pussy out if you take this far enough. You seem pretty intelligent, and I expect the Jobcentre see you as a statistic so assume you're a moron who can't get a job because... you're a lazy moron.
As for papers, no idea, don't even know if bad publicity will achieve anything since they're so closely allied with the government (I wouldn't be surprised if the Conservatives are the ones to blame by advising the Jobcentre to sanction everyone possible to save money).
It is a system where the chavs get paid weekly, but the honest can get punished. It's completely in reverse, and it's embarrassing for us as a country.
I don't know the success rate of appeals, but it can go on for a while and in an unbiased setting the Jobcentre would get done.
I don't really have any advice, sorry, but you do seem to be on the right track. It can all become a bit of a mess with financial terms flying about, things to sign and it relies on nobody involved at the Jobcentre makes a mistake with the info they're given. Good luck.
I also hope you're not getting all depressed over it. I don't know your circumstances, but just imagine. It's like finding love, it'll come at some point, out of nowhere. Sometimes it won't work out, but eventually if you're smart enough and you work, this will just be a memory while you live in your home with a present and a future.
Hold on to that thought, that potential.