I might have some bad news for.
The 3rd year of the warranty is not covered by Fiat, but by another company who are very strict, and don't like paying out. They require the dealer to fill out a claim form to include mileage and service history, which is then e-mailed to the warranty company.
Like yourself, My missus's 2001 Punto 1.2 GO! was in its 3rd year of warranty, when it went in for a 36,000 mile service and to fix a leaking rocker cover gasket a few months back.
They refused to repair the leaking gasket under warranty as it had done 38,000 miles.
Of course it may just be my dealer, Meredith & Kirkham of Colwyn Bay, who are talking crap as they are absolutely useless, but you're dealer may be more helpful.
And as for the service book, they probbaly have the car's details on computer so they know the date it was last serviced.
Here's another example of the how bad the 3rd year warranty is. My HGT went in for it's 24,000 mile service (it had only done 20,000 miles)and to investigate a noisy heat shield and a faulty variator.
They immediately refused to to repair the heat shield as it could have been damaged by a stone or something, now bear in mind that the heat shield in question is located behind the radiator where it is impossible for a stone to ever reach. They did agree to replace the variator, but it took three and a half weeks for them to give me a decision.
I'm sorry to be the bearer of bad news, but I would try to reason with them that 4,000 miles over service would not cause the ECU to malfunction.
Anyway, wait and see, you never know they may overlook the service record.
Good luck and keep us posted as to how you got on.
Evan
Broom Yellow 2002 HGT.......mmmmmmmmm! nice.