It was £150 and change including VAT, which is still less than 12 months' warranty would have been (but more than 6 months'). However, there's some doubt as to whether that would have been covered. If it falls under wear and tear then despite what they say about "avoiding dispute and annoyance on your part" they would not have covered it because the policy would have been less than 90 days old. (It doesn't seem possible to get an actual list of what is covered without doing a quote, which I'm not doing again. Of course I still have the booklet somewhere at home, but it's only for the more expensive ExtraCare option.)
The arrangement by which I bought this car falls somewhere in the gap between a private sale and a trade sale. It was bought from a trader, but he only had the car because he had taken it in part exchange and he wasn't offering any value added services. (In fact he claimed that he was offering it for sale at the same price he had effectively given for it in the part-exchange deal, though I suppose I don't have to take that claim at face value. I think that if true this means he offered too much for it.) He hadn't even had the car thoroughly checked out beyond doing an HPI check, inspecting the body, and driving it around for a while. In effect therefore I was buying the car off its previous owner. I had had the option of taking out a warranty at my own expense, and declined it.
Should your faithful correspondent be liable for the repair?
Yes. Tough break. There was no way anyone could know that would happen, and it will probably be fine from now on.
Probably. Send a polite letter to the trader to explain what has gone on.
No. Demand recompense from the trader.
No. Sue the credit card company.
Some other answer.