Vehicle faulty within 6 months of purchase but dealer won't refund repair

[Guest]

I purchased a second hand vehicle from a dealer and within 6 months the engine ceased in the midst of traffic at a traffic circle. I had the car towed to my garage who said it may be a faulty fuel injector but that the engine would need re-assembling or I would need to replace it. It was slightly cheaper to replace with a second and engine so I did so. I notified the garage I bought the vehicle from as they had sold me a warranty at the same time of purchase. The warranty inspector checked the vehicle over at my garage and his report said it appeared to be drive on damage (only drove 3,500miles in the 5 month period of ownership) as there would have been misfires and indication of issues prior to the engine ceasing and attending to the condition sooner would have resulted in the fault not happening. The report said the condition would not have been present at the point of policy inception, in light of the time elapsed since policy inception, which he believed to be over 6 months (which is incorrect). Dealer rejected my request for a refund for repair as they said I had owned the car for 6 months. I pointed out the purchase was 5 months before the issue and that the warranty report was not sufficient alone to say that the car was not fit for purpose, they have to prove that the fault wasn't present upon purchase. They have stated they are not responsible for refunding me because of the warranty report findings as set out above. I did not realise at the time I should have had the car towed to the dealership so I realise I may have shot myself in the foot slightly. My understanding of the CRA 2015 is that the fault is deemed present though because it occurred within 6 months of purchase. The dealer has said given the fault is not latent or inherent they will not pay £169 investigation fee for my specialist garage to investigate the engine failure nor the £2,000 plus VAT (excluding labour costs as I did not request this) for a replacement engine. Can the dealer rely on the warranty report as evidence that they did not sell me a car that was not safe or fit for purpose. I should say that at no time did I experience any issues with the engine nor did any warning lights appear until the engine gave in. Any guidance on what to do next would be much appreciated!!