This is really tricky. In my view, either you buy privately 'as seen' and you know you are on your own, relying on your 'car' judgement, and your experience. Remember when you buy such a private car, you are buying 50% the car and 50% the seller and paperwork. Fair enough, as you will get it cheaper, buying private, but there is little comeback.
Or, go to dealer and pay more, for a dealers full services. You seem to be half way. If the stamp on the contract says what it does, one normally would see bells ringing...... i.e. trouble. Did you pay the £20+ to get an HPI report, or run it through 'Check an old MOT history' on the web.
I know this sounds patronising, but he knows more about the car than you (asymmetric information), so you must protect yourself. It seems like you thought it was a bit 'iffy' and got a £1,000 off ( an unusually large sum). Big money. You took it almost on trade terms, i.e. taken as seen , no comeback. Why did you agree to that stamp? It rings trouble. I fear a small claims court would say it is clear on what terms you purchased.
By all means try to get the dealer to cough up, but I would not put my pension on it.