No, they are wrong. You have statutory consumer rights here. After 30 days of ownership but under 6 months - they have one chance to repair the vehicle at their cost (they cannot ask you to "go halves" on the cost). The consumer law is clear in this regard so they are trying to fob you off, as is usual with some dealerships.
It would be helpful to summarise the law for you.
If you purchase a used vehicle from a dealer you have rights under the Consumer Rights Act 2015, which states that the vehicle has to be:
· of satisfactory quality in that it must not be faulty or damaged when received
· as described in that the vehicle must match any description given at the time of purchase
· fit for purpose in that it should be fit for the purpose intended
If the vehicle does not meet the above then you will be able to enforce your rights and sue for your money if the dealer refuses to give you the money back. The claim would not be against the manufacturer as they will only be responsible under their warranty which came with the vehicle.
There is no protection against fair wear and tear, misuse or accidental damage, faults that were pointed out at the time of sale or if you change your mind and no longer want the goods. The age and value of the vehicle will also be relevant and the expectations of older vehicles will certainly be lower.
The rights against the seller are:
1. Reject the vehicle and request a refund which must be done within 30 days of purchase or delivery. The faults are presumed to be there from the outset unless the dealer can prove otherwise.
2. Repair or replacement and this can be done within the first 30 days or after, if that deadline has passed and a rejection is no longer possible. If a repair is not possible or has failed, the vehicle can still be rejected for a refund, or if you want to keep it, you can ask for a price reduction. If asking for a refund or a replacement, the current value of the vehicle will be used, taking into account any depreciation in value for usage by the buyer since purchase.
If the repair would put you are considerable inconvenience - e.g. taking several weeks, or meaning you have to travel a long distance to them, then you can demand a refund. They are only allowed one chance at the repair - if that fails, you are due a refund. As I say, to ask you for half of the repair costs is illegal. And the warranty does not affect your statutory consumer rights so they are wrong to demand this from you.
You need to contact the dealer and advise them they have one chance to repair at their cost and if they refuse, then I would advise that you report them to Trading Standards (https://www.nationaltradingstandards.uk/contact/) and commence legal proceedings via the money claim online site (www.moneyclaim.gov.uk) for your money back.
Before you do this however you should send them a 14 day letter before action to warn them of your intentions, to allow them to avoid litigation and resolve this dispute. I attach a template letter for you to use which will hopefully resolve the dispute.