It's not up to their insurers to write it off. They have an obligation to return your vehicle to how it was before their client damaged it. I've heard of cases where people in your situation have had the other insurance company cover the cost of any repairs that were required. It does not matter that the cost of repair outweighs the market value of the car. Although I'm not sure what the extensive damage means in this situation.
I wouldn't be accepting any cash settlement from them. You want your car returned (or equivalent) in the condition it was in before their client crashed into it. Maybe speak to CAB about this, or your own insurers.
If you are in the RAC or AA (and as you drive FIATs why would you not be?) they offer free legal advice re such matters,. might be worth a call.