Just keep in mind there's no guarantee there won't be negative publicity after the goodwill job. A used engine could be asking for trouble too. In any event good luck!!
MY FIL was an HVAC guy and automotive AC expert. He had a tank full of nitrogen to find AC leaks. He said it was one of the easiest and least expensive ways to track down AC leaks.
This is likely going to end up a no win situation. Giving the customer money is admitting guilt, people don't handout money for no good reason. Here's a few possibilities of what is likely to happen. Even if the OP were to hand money out, odds are after the payoff the customer is going to say I...
For those mentioning the shop's insurance. Even with insurance they might not cover the repair. IIRC the customer would have to sue, depending on how high the deductible is the insurance company will only pay above the amount of the deductible, and most deductibles are pretty high. Then they...
Apparently even with all the idiots on the road it is enough of an issue for the NHTSA to address and make the news this morning. Regarding idiots there are plenty of them driving all makes and models, so there's one point we might agree on.