Just as a punter who has dabbled in buying and selling, but by no means expert, I think you will face a couple of difficulties making some of your points prevail
- generally garages & auctions do not own cars, and their statement of previous ownership reflects what the current/actual (DVLA recorded) owner would say, not what the car will become once sold. I don't know for sure if that is ethical, but I never questioned it, maybe because I was aware it happens and I always check the log before purchase (to check lots of facts)
- I also think there is a standard set of faults categorised a "major" that generally pertain to mechanical propulsion - engine, gearbox, drives type of stuff. Although the problem you mention, could be expensive to fix (or even diagnose), it may not be defined as "major". You'd probably find the definition of "major" in the context of your car in the guarantee paperwork.
- however, even if the fault is not "major", they need to put it right. For this, you'd need to comply with their request to "check & rectify". I think they get a set number of chances to get it right, and if they fail to do so, you then have a valid case. If you don't give them the opportunity to fix (in the absence of a "major" fault), you'll struggle to win the day. As I mentioned, I am not expert, but to my knowledge, that is the established process. Independent reports etc - I think you are wasting your time and money, and lets face it, anyone can get a report to say anything
- you have to consider that the fault may be very simple, and second hand cars always have minor issues - you don't know, nobody knows, but it may simply be a loose connection
- if I were in your shoes, I'd seek out the previous owner, and ask him two questions (i) did the fault exist previously, and (ii) did the seller know. If he obliged you with these answers, I'd hold them up my sleeve for a rainy day. In parallel, I'd let the seller do his best to fix.
Just amateur advice, hope it helps somewhat - someone else may know better. Good luck