I'm sorry to hear you've got yet more problems.
Sadly all of this is exactly what is likely to happen if a FIRE engine is ever run low on coolant.
It would not surprise me to learn that this engine had overheated in the past, been patched up and then traded in/sold on. Unfortunately you've inherited the problem.
The difficult bit is advising you what to do next. I'd like to see the selling garage taking some responsibility here, but unless you bought it from a franchised main FIAT dealer, I think it will be difficult, frustrating and time consuming to get any real satisfaction from them. If you did buy from a franchised main dealer, then FIAT Customer Service is your next point of contact.
IMO you should be looking at a solution which leaves you not owning this car; whatever is done now, I think it's going to be a real lemon from here on. Pragmatically, you might consider selling it "as is", and putting whatever you get (which could be more than you might think) toward something else and starting again. Perhaps a brand new car on a PCP. The 500 has historically been a good car to run on a PCP, since high residuals have kept monthly payments low.
Don't enter into any agreement which runs for longer than the warranty on the car.
Anyone contemplating rebuilding a seriously overheated FIRE would probably do better to source a secondhand engine; they can sell for less than you might think, and IIRC @
typecastboy actually threw one away after selling the other stuff of value on the car.
I've said this frequently, but IMO unless you're a highly competent home mechanic, if you can't afford the payments on a new one, you likely won't be able to afford the bills (and depreciation) that come with a used one.
And good luck; I hope you do find the best outcome possible. I'm sure all of us here will do what we can to help, and we'll be alongside you on your journey, whatever course of action you choose to take.
