Antiquities of the Jews by Josephus contributes three things to our understanding of our conjectured trial.
First, the Josephus passage demonstrates that one aspect of Philip the Tetrarch’s work in his territory was judicial. In this role he can be understood as a traveling circuit judge.
He settled conflicts between tribes and ethnic groups. He also judged all capital cases. Local matters were settled within a tribe itself, according to the laws and customs of the tribe.
Philip the Tetrarch was the only person in his jurisdiction who was automatically authorized to pronounce judgment in a capital case.
Second, Philip’s result was the same result expected in any Roman court, the determination of guilt or Innocence.
The fact of capital decisions (death penalty cases) coincides with the Talmud and with Luke.
In the Talmud, all three of the passages cited (click here) relate to capital cases involving sexual transgression. Each Talmud section features a legal expert justifying the destruction of something not ordinarily subject to Jewish law (the inanimate objects and a beast or ox). In Shabbat 6:7 the judge is labeled Antigonus the Prince.
In Luke 17:34-35, a capital determination seems implicit in the twice-used word “seized”. One man and one woman are “seized”, that is officially taken into custody, presumably for execution.
The key things to note in the Josephus passage are 1) Philip’s judicial role, 2) that he adjudicated whenever and wherever his presence was required, and 3) that justice required his determination of guilt or innocence under the law.