
I would say so - his job was to 1/ Identify the fault and 2/Repair that fault.
He only did part 1 of that job, part two he failed on so he owes you at LEAST a complete refund on the part (because the part he got and replaced was not necessary and was his error) and a partial refund on the labour (because he did correctly diagnose the fault).
I would set this out in writing - keep it factual and don't put any emotion into it at all. Be clear about what you are asking for and why, and then state that you will follow up with contact to trading standards and potentially small claims court if necessary.
If he is saying the info isn't in the manual that he had access to, and it is, then quote that also - all goes in your favour. He has messed up, and really, why should you have to pay the full price for that?