I hear you and agree that the umpire shouldn't be liable. However, a lawyer could argue that the umpire should be taking note of the bats used by each batter. Again, I don't agree, but in the court of law, who knows.
I am sitting in Zoom court waiting for my client's case to be called while i write this. I have been asked various versions of this question by my friends over the years.
This kind of case is only going to occur if there a very significant and/or tragic injuries. EAJugalo is very smart to get supplemental insurance to cover him for his umping. Often the most important part of insurance is not the duty to pay the claim, but the duty to defend the policy holder. While the umpire would most likely be found not liable, it could cost tens of thousands of dollars to "win."
Under the original scenario, there is not the correct organization certification, this is a very tough case to prove. You would have to prove that if the bat met the proper organization certification, the performance of the bat would have been sufficiently different so that the injury would not have occurred if the bat had the correct certification. You would need an expert witness to write a report and testify about the different level of ball speed, performance etc. The defense would most likely have a report indicating that the performance and ball speed would not have been sufficiently different to cause the injury. The umpire would take the position that he checked the bats before the game. He can't be expected to know what bats have what certifications by sight during a game. The case against the batter would be difficult in this scenario as well.
If the situation was one that the bat had been deemed illegal that may be a slightly easier case. Again you would need to show that the performance of the illegal bat was sufficiently different. The plaintiffs argument would be that there is a list of bats that has been published of which bats are deemed illegal. These bats are well known and a reasonable umpire should know the list and should be able to spot a bat on the list during the game. The umpire has a duty to be familiar with that list. I believe there is some validity to this argument. Again it is a difficult case against the umpire but much better than the certification scenario. The case against the batter would be similar but much easier. There is a widely known list of illegal bats and you are using this bat because you know it is "hot."
If the situation was one were the bat was shaved, doctored or painted over, I would be hard pressed to create a theory against the umpire. Assuming the umpire was unaware. However, it would be a very strong case against the batter. In this scenario you could show that the batter knew he was getting an unfair advantage. The defense that the injured player "assumed the risk" would not work. The defensive player did not agree to play against doctored (ie fraudulent) bats.
Final situation is this, Batter A uses his teammate B's doctored bat and causes injury. Batter A did not know the bat was doctored. Batter A would not have liability. However, B could be found liable for the injury caused by A.