Something doesn't sound right here. In fact, many things don't sound right.
BNR has tuned over 500 cars, mine included, and most of those have been 1.4L Turbo engines. The OP is the first to report an engine failure on that tune.
My vehicle has been run short trips to redline making full boost every time I take the car out for a spin, for any purpose whatsoever, and it has made that power reliably for the last two years or approximately 25,000 miles. Those were some very hard driven 25,000 miles. The OP's engine somehow "blew" with only 4,500 miles. It doesn't add up.
The OP states that they installed the factory tune back on. How then did GM find the tune and claim it voided the warranty? All GM could have found was that the ECU was programmed at some point, but they have no way to prove that the programming caused a failure.
The dealership offered to buy back the extended warranty, the car itself, and to pay off the loan of $9,000, and there is no indication from the OP that this would be part of a new vehicle purchase. Why would the dealer make such a generous offer if, as ChevyGuy pointed out, that is basically the value of the vehicle?
The dealer said the engine blew, and said he needed a new engine. There is nothing a tune can cause that will require someone to need an entirely new engine short of extreme detonation, and BNR's tunes have safety parameters in place specifically designed to dial back power, even throw the car into limp mode before the vehicle even gets close to that point. If the pistons blew, which I've seen occasionally happen on these vehicles, tuned or not, the pistons would be replaced, not the whole engine, and it wouldn't cost $4,000 to do it.
The cost of the service to be performed is absolutely asinine. A used 1.4L Turbo goes for $500, maybe $1,000 from the most expensive junkyard in town. The remaining $3000 is for what? 24 solid hours of service? It takes one day to pull and replace an engine, not three. The math doesn't add up.
A lot of things don't add up here. To the OP: did you datalog your tune like you were supposed to, install any updates BNR sent, and send datalogs for any update that was needed?