Hello and welcome to the forum.
So very sorry to hear it's in such sad circumstances. I can understand how gutted you must feel.
Unfortunately the garage which currently has possession of the car has you over the proverbial barrel. Either you pay them to fix it, or you pay them for the work already done and have the car trailered away someplace else. I don't see how they could reasonably refuse to fit a replacement 'box supplied by yourself, but they'll likely load the labour charge and obviously won't then warrant the replacement 'box.
Most folks in similar circumstances have reported that Fiat will do little or nothing to help; pursuing that is more likely to lead to more frustration and anger than to achieve anything useful.
DMF's are a wear and tear item, but sadly on the TA they can cause quite a bit of collateral damage when they let go. Stories like this just reinforce why I'd not personally want to own a TA once out of warranty.
Pragmatically, I'd get it fixed as cheaply as possible and then quickly trade it in. Who knows what other components have been stressed? Paying £2000 and then having the inevitable fight for restitution if the repair doesn't last isn't a place you want to be.
If there's a lesson here for other TA owners, it's not to ignore any symptoms of excessive DMF wear; fixing it before it breaks something else is likely to prove considerably cheaper than waiting until it's failed completely.
