Thanks for your question.
Yes, if there is a genuine fault with the car you can recover either the money for the car or damages in respect of the fault. Stop dealing on the phone with them, only in writing so that there is documentary evidence that you can use to assist you if you make a claim.
The law states that up until 6 months after the sale of the vehicle, it's up to the dealer to prove that the fault wasn't there when he sold it to the purchaser. This is called the "Reversed Burden of Proof". They will know this and you can cite it to them if they claim "it's your fault"
Go back to them, state that the car is not of a satisfactory quality under Sale of Goods Act 1979 and that you require that they repaired, they can try and fob it off to Fiat Customer Services but it's not their problem - it's the dealer's problem if there's a problem with the car and Fiat know that.
You are entitled to a replacement car during the dispute. If it can't be repaired and the disrepair is so serious that it deprives you of the whole, or substantially the whole, of the benefit of the car then you are entitled to either a replacement car or a full refund because of the dealers repudiators breach of contract. This is essentially a breach which is so serious that it entitles you to treat the contract as terminated.
You are not exposed by taking the car from the forecourt to an independent engineer, your rights under the contract subsist whether you take it or not.
At some point you may wish to threaten them If they are not compliant you should tell them you are prepared to make an application in the small claims Court and will also contact Trading Standards, http://www.tradingstandards.gov.uk/ , and Retail Motor Industry Federation (if they are a member, http://www.rmif.co.uk/ .
If you get nowhere corresponding you may wish to ask a local solicitor to write a one-off letter before action to show your serious, this is fairly inexpensive. If you have to issue and the amount of money you are claiming is less than £5000.00 then you can claim it yourself through http://www.moneyclaim.gov.uk/. If is is greater than that they you will have to issue on the Court's fast track and you will be able to claim for legal fees, interest and other costs.
If this has been useful please kindly click accept so that I may be rewarded for my efforts. It will be gratefully received and you will be free to ask follow up questions.