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 of clarified what counts as substantive law, and what counts as procedural law. Punitive damage limitations generally count as substantive law under this test.
So, the question of which state's law to apply to punitive damages caps in a diversity case brought in a U.S. District Court in Texas is precisely the same as the question faces by a Texas state court regarding which state's law to apply.
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.
Historically, there were rigid rules that provided that in particular kinds of cases, a particular state's law would apply. Some of those historical rules continue to have full force.
For example, the substantive real property law of the state where real property is located almost always applies.
But, in other cases, a more vague modern rule for choice of law applies. This is the rule that the law of the state with the most significant relationship to the issue to be decided should apply. Different issues in the same case may end up being governed by different laws as a result.
Among the factors to be considered in applying a most significant relationship test are:
The place where the event giving rise to the claim occurred.
The interest of the state where the person who was harmed resides in controlling the amount of compensation that the person harmed received.
The interest of the state where the person who caused the harm resides in encouraging or discouraging business activity that may hurt others.
The interest of the state where the insurance company paying some or all of the claim is located in not driving up insurance premiums.
The inquiry in this multi-factor balancing test is usually extremely fact intensive.
The practical effect of the most significant relationship test has been to make the ultimate decision less predictable and to make it more likely that a judge will decide that the forum state's law will apply. Under the historical rule, about two-thirds of requests to apply a non-forum state's law were granted. Under the modern rule, about one-third of such requests are granted.
A law review article from 1987 spells out the choice of law rules in Texas at the time which have probably become somewhat looser since that time that it was written. The article is James P. George, Choice of Law Outline for Texas Courts, 18 Tex. Tech L. Rev. 785 (1987). Available at: https://scholarship.law.tamu.edu/facscholar/255
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:
Erie has gone in a newer and even more complicated direction than the previous controlling cases, and that instead of selecting either federal or state law for a case, the federal court may be required to somehow blend federal and state law, depending on the issue. This is quite frustrating for those who wish to have a black-letter rule that will point them to the answer. However, the possibility of blending in Erie does not open up an infinitude of possibilities.
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.