The fact that the car had an MOT only says that it was roadworthy on the date of the test. It may not necessarily have been roadworthy the following day.
However depending on what the fault is that manifest itself eventually, it may be that the MOT was defective. I will say that this is not common.
Under the Sale of Goods Act the dealer must sell a vehicle which is of satisfactory quality and fit for purpose. It must remain so for a reasonable period of time in the courts are found that 12 months is reasonable although that varies depending on the age of the vehicle, the mileage, any alleged warranty which comes with it, and the amount paid and whether any faults of being disclosed.
It appears that your daughter purchased a 10-year-old car for a substantial amount of money which actually included a warranty. I don’t what the terms of the warranty are but if these things that have now turned out to be problematical are covered under the warranty are not specifically excluded, then whoever provides the warranty (it could be an insurance company or it could be the dealer itself) is liable for those costs.
It appears that this item isn’t actually covered under the terms of the warranty and therefore you can’t expect them to replace it under the terms of the warranty.
Almost 2 months after the MOT I also don’t think that you can rely on a defective MOT.
It therefore comes down to whether considering the time that your daughter has had the vehicle, these faults existed at the time it was sold. That would come down to whether you will get evidence which confirms that these faults were there at the time the vehicle was sold (it wasn’t of satisfactory quality at that time) and that they have only just manifested themselves.
If I are reluctant to issue you with a copy of the original emission certificate from the MoT, that is remarkably suspect and I think you should insist. Tell them that unless they let you have it, you will refer this to VOSA as a complaint in respect of a fraudulent MOT.
You ask whether you can get the work done on the car with no cost to you.
After several weeks these problems shouldn’t manifest themselves and I think it’s borderline as to whether they existed at the time of sale. It wouldn’t put me off pursuing them through the Small Claims Court is necessary. Hopefully, if you do issue proceedings or at least threaten them, they would take a more pragmatic view.
Even if I told you that the dealer was completely liable for this (I think it’s probably 60% in favour of you) actually getting them to do the work and being liable for it are two different things altogether. Unfortunately, the motor trade has a habit of being happy to sell new cars and to sell used cars but then lose interest when their profit get eaten into because they have to do warranty work.
Can I clarify anything for you?