Small Claims court - right to reject a vehicle

[Juno22]

In October I purchased a vehicle from a trader for just shy of 6K, and within 5 days I noticed an oil leak. I messaged the dealer to try and fix the issue as I knew they offered a 30 day gearbox and engine warranty (as most do). I took the car to them and within a day they said they had fixed the issue, although no proof of work was given to me and no garage invoice but a text message saying they'd fixed the sump seal and topped it up with oil. Anyway, in early November a check engine light came on. At this point I was beginning to loose faith in the car. A quick read of codes informed me that there were a couple of issues to do with engine oil pressure sensor and mass air flow sensor. At this point I decided I would reject the car on the grounds of it not being fit for purpose (I conduct long motor way journeys of up to 3-4 hours each way on weekends) and not as described (Car was described on the advert as being in 'great condition'). The vehicle is 8 years old and 120K on the clock but had just had a squeaky clean MOT. Both these faults happened within 30 days of purchase.

I messaged the dealer to let them know I was going to reject the car for a refund and promptly followed up with a formal letter to the company address, stating that I wanted an acknowledgement of them receiving the letter (I used royal mail signed for special delivery so I know they have received the letter) and that they had two weeks to resolve the issue. It's been 5 days since they received the letter and so far have had no acknowledgement of the letter and no response about resolving the matter. Interestingly, I can't find any information on Companies House about the business name on the invoice, but the business account I transferred the money to for the purchase appears to be affiliated to the traders private property, and another different company name is affiliated with the address on the invoice.

Could anyone please offer some advice on next steps on take and whether they think this is worth pursuing in the small claims court?

[Pezza54]

A company can trade in a name that isn't the company name registered at Companies House. However the registered company name, address and company number must appear on an invoice. Check the bottom of the invoice for small print. This is so a potential claimant knows the right company name and address to include on court forms.
If you should decide to make a court claim the defendant will be the company registered at Companies House that sold you the car.
When you rejected the car you should have told the dealer the car is available for collection at a suitable time to be arranged.
You should not drive the car.
Although you didn't mention the Consumer Rights Act 2015 in your thread you appear to be aware of this legislation
If the dealer refuses to let you reject the car you could:

Make a court claim against the dealer for a full refund plus damages (alternative transport costs, garage report). A claim on the small claims track is currently taking 9-12 months. You will have to park the car safely for this period of time.
or
Arrange for the repair work to be carried out at a reliable local garage and then make a court claim against the dealer for the repair cost plus damages.
or
See if the dealer has a suitable replacement car and you can reach agreement on the financial difference.

Hopefully the dealer will let you reject the car, or negotiate on a suitable replacement to avoid going to court.