Why did the Jews say they had no law to kill anyone at Jesus's trial, yet claimed the right to stone the adulteress?

Because at Jesus's trial they were accusing Him of high treason against Caesar, and a political execution of that kind was reserved to the Romans; once you take that route, the decision to execute is left with Rome. Pilate, seeing no treason, said he was giving Jesus back to them, but as prefect of the region he could not allow the Jews to carry out an execution, or he would be removed the moment Rome heard he had let them break the law on executions. The stoning of an adulteress, by contrast, was the Mosaic law, given about fifteen hundred years earlier when Israel was its own theocracy with no Romans present; then they could stone anyone their leaders under Moses judged guilty. But in Pilate's day the Jews were not allowed to do that. That is the reason for the distinction: it is the difference between a Roman political execution and the old Mosaic penalty they could no longer carry out under Roman rule.