CRA/SRA used car question - waterpump & cam cover leak

[byepolo]

Hi all.

I know this has probably been asked before, but I'm still not sure where I stand.

I bought a used car in Feb 2024. Today, I took it to a garage for a full service because the dealer/trader claimed it was done, but his mechanic just "forgot" to reset it and stamp the book.

Anyway, I got the car back, and the garage says the water pump is leaking, which I suspected because I've had to top it up twice in the past two months, and the cam cover is leaking. They found oil in cylinder 4, which isn't a priority, but it will be a £1k+ job in a year's time.

Do I have a leg to stand on to ask the trader to get it repaired since it falls under SRA/CRA 2015 as it was purchased less than six months ago? I'd like to note that I've driven 4k miles on it, so they might claim it as wear and tear, but it's up to them to prove that, right?

From what I've gathered from old posts, if the dealer is going to be difficult about it:

-Send a written letter, wait five days for a response. If there's no response or refusal, get the work done, and go to small claims court.
- Same as above but a private solicitor - probably not worth £2k in fees though?
-There's a third-party warranty with AA, but they probably won't cover this as it's a BS warranty anyway.
-Lesson learned. Get it done and just forget about it?

Thank you!