This is a horrible position to be in, you have my sympathy. My decision to buy a Honda this time around was partly based on their reputation so this kind of story is a bit worrying. It is of course up to you whether you wish to pursue this, but do please make sure you understand your rights. For example, the following isn't quite correct:
As it's beyond 6 months we have no right to reject it, it was reported within 6 months but we had no evidence
The issue (under the Consumer Rights Act) is whether the fault was there at the time of purchase. From the Which? link, if you're within 6 months you don't need proof, it's up to the retailer to prove it wasn't. If you reported it within 6 months, with video evidence, that would indicate to any normal person that the fault did exist very early in the car's life. After 6 months it's more complicated, the onus of proof is on you so you may need to get an independent report, but that doesn't mean you don't have a valid claim against the dealer ... and you have 6 years to make that claim.
Out of interest, if they said you weren't starting it correctly, what advice did they give you about the correct procedure ?