Dont worry about it,your annoyed & so would anyone be.I would go down the "you sold me a car with fault(s) therefore you should repair them" route first. If they wont then go to customer services,be polite & simply state you were sold a car from Fiat agent with faults,pictures etc are important here.
Let the garage & Fiat fight it out between them as to whom is paying.........it should not be you.The basis of this is that if you brought a 6 year old car the garage is obliged to repair any defects (normally within a 4 week period) which they have to so why not a 2 year old car?
My view on 3rd year warranty is that it is confusing to the customer (and much of the time the garage) & should be done away with, in its place a booklet should be provided clearly showing what is/what is not covered.
One point is the garage should in any case be contacting Fiat or area manager to seek authorisation to carry out the faults you have,it should not be left to the customer to get stressed out trying to get a £6k (?) car repaired after recently buying it.