Short Answer
At the level of generality presented in the question, it is impossible to know with any meaningful level of comfort, which state's laws regarding punitive damages caps will apply, even if there are prior precedents addressing a similar issue, since the determination is so dependent upon a rich set of relevant facts.
Long Answer
Does Federal Or State Law Apply?
In a diversity case, under what is known as the Erie doctrine, after the U.S. Supreme Court case that established it, a federal district court sitting in diversity applies the substantive law of the state in which it is located (including its choice of law rules) and federal procedural law.
A variety of subsequent cases ofhave clarified (or at least made more specific) what counts as substantive law, and what counts as procedural law. PunitiveA 2018 law review article (open access) uses a five page long flow chart to explain the rule. A simplified partial page flowchart regarding whether state or federal law applies can be found here.
To cut a long analysis short, punitive damage limitations generally count as substantive law under this test.
Choice Of Law Analysis At The State Level
In general, a state court (or a federal court sitting in diversity) applies the law of the state in which it is located unless a party argues that under choice of law principles that another state's law should apply, and that party shows that the law of the state whose law is proposed is different from the law of the state where the case is being tried.
It also observes in an analysis that should be taken with a grain of salt, because it comes from a law firm that defends companies that are usually defendants in product liability cases rather than plaintiffs and is not written by a neutral party:
So, at the level of generality presented in the question, it is impossible to know with any meaningful level of comfort, which state's laws regarding punitive damages caps will apply, even if there are prior precedents addressing a similar issue, since the determination is so dependent upon a rich set of relevant facts.
To some extent, these choice of law principles apply even in cases where there are arguably procedural rules that apply, rather than merely blindly following a federal procedure and state substantive distinction. Wikipedia's entry on the Erie doctrine concludes in part by stating:
A 2018 law review article (open access) uses a five page long flow chart to explain the rule. A simplified partial page flowchart regarding whether state or federal law applies can be found here. But even in the context here where it is clear that state law rather than federal law applies, the determination of which state's law applies is itself involved and is frustratingly indeterminate.