Sanhedrin (81b) used the kipa when they thought the guy was guilty but there were technical reasons that they couldn’t give the death penalty. Kipa was a locked room where they’d feed him barley to make his stomach explode. They may use the kipah if there was aidus. The important thing to remember about kipa is that it is “self inflicted”.
There is a Ran in Droshas HaRan on Parshas Shoftim who says that aside from the system of beis din which was limited in its ability to punish, there would also be a legal system directed by the king or other authority who would have the ability to punish criminals without the halachic restrictions; for otherwise there would be anarchy. Beis Din itself had extra-judicial powers it could use when it deemed there was an emergency situation. This extra-judicial power could even allow it to execute someone even if normative halacha didn’t impose it.
Sanhedrin 37 tells of a man who saw someone chasing down another person into the forest and then saw him standing over the dead body, holding a bloody sword. And yet, this man could not bring the murderer to Beis Din even though it was obvious that he committed the crime, since he was only one witness, and technically, there was only circumstantial evidence.