Approach the dealer anyway. Under consumer law, any fault arising within six months of the date of sale is held to have been present at the time of sale unless the vendor can prove it was not. These are your rights under the Consumer Rights act 2015. I recommend you do a little web based research around this law, then contact your dealer and see what they say. You may find that they will deal with this without a struggle.
Hopefully, you may be entitled to have this dealt with, without footing the bill. This quote is lifted from the What Car website on your consumer rights:
If a problem is found after 30 days, but within six months of purchase, you can request a repair or a replacement vehicle. The onus is on the seller of the car to prove the fault wasn’t present when it was sold; if they can, and you’re likely to have known about it, you won’t get a refund.
When it comes to repairs, the dealer has one chance to fix the fault. If the fix doesn’t work, you are entitled to a refund. You won’t get the full purchase price back, though. Any usage of the car will be taken into account and the refund will be adjusted accordingly.