The law is quite clear here, in that if there is a fault reported with a vehicle you sell as a trader then you first of all have the right to repair it before the customer gets the work done elsewhere. The grey area here is that you offered to provide the parts, but it's not clear if you offered to get the car repaired yourself. If you didn't offer to do so and told the customer as much, then they are within their rights to get the car repaired elsewhere. Your best bet would be to contact motor trade law specialists Lawgistics, who offer a free initial consultation and will be able to determine if the customer has a legitimate case against you – www.lawgistics.co.uk