Q. Does any historical precedent suggest that the Romans would tolerate Paul's crusade in Acts 9:1-2, because he got a Jewish priest's approval?
[Act 9:1-2 NASB95] [1] Now Saul, still breathing threats and murder against the disciples of the Lord, went to the high priest, [2] and asked for letters from him to the synagogues at Damascus, so that if he found any belonging to the Way, both men and women, he might bring them bound to Jerusalem.
I wonder if the precedent was Ezra ch7 vv25-26, where Ezra has delegated authority over all those of his own religion in the province "Beyond the River" (which is, of course, named from the Persian viewpoint and is roughly the equivalent of the Roman Syria).
The Sanhedrin were claiming a similar authority over Paul himself once he had been arrested, complaining that he was an agitator "among all the Jews throughout the world" (Acts ch24 v5).
Probably. The Romans allowed the Temple guards to arrest Jesus in the Gospels and according to the Jewish Encyclopedia Jewish religious courts also maintained local police forces. Bringing people back in chains to Jerusalem is most problematic part of the scenario.
Pre-Roman times
There are many terms in the Bible which have been translated to denote magistrates or police officers; but the correctness of the translation is questioned.... The Deuteronomic code (Deut. xvi. 18) enjoins the appointment of "shoṭerim"(LXX. γραμματοεισαγωγεὶς...) whose duty it was to execute the decisions of the court. The frequent mention of the shoṭerim together with the judges (Deut. xvi. 18; Josh. viii. 33, xxiii. 2, xxiv. 1; I Chron. xxiii. 4, xxvi. 29), or with the elders of the community (Num. xi. 16; Deut. xxix. 9, xxxi. 28) who acted as judges in earlier times, would seem to indicate that these officials were attached to the courts of justice, and held themselves in readiness to execute the orders of the officiating judge.
Second Temple times
The Temple had a police force of its own, most of its officers being Levites. These were the gatekeepers ("sho'arim"; I Chron. ix. 17, 24-27, xxvi. 12-18), the watchmen that guarded the entrance to the Temple mount, and those that had charge of the cleaning of its precincts... The Mishnah (Ket. xiii. 1) mentions two judges of "gezerot" (lit. "prohibitions," "decrees"), who were in Jerusalem during the latter part of the second commonwealth... it is safe to assume that the functions of these judges were similar to those of modern police magistrates...
The Second Jewish Commonwealth here refers to the period between the last third of the sixth century B.C.E. to the destruction of the Second Temple by the Romans in 70 CE. So the above paragraph includes the time that the OP asks about. Also, according to Joshua Kulp the gezerot should be understood as "prohibitions created to prevent a situation which might cause a person to transgress."
Conclusion: If Paul were employed as the agent of the high priest or a religious court as a judge of gezerot, then there is possible precedent that his activities would be allowed by Roman authorities. The report that Paul "went to the high priest and asked him for letters to the synagogues in Damascus" could be Luke's way of expressing Paul's getting hired as judge of gezerot assigned to Damascus. On the other hand, there is no known evidence outside of the NT that the high priest or any Jewish court ordered that Jewish Christians be arrested and brought back for trial in Jerusalem. Normally violations of gezerot would be handled locally. Nevertheless if the letters of the high priest constituted a decree aimed at preventing a major transgression such as blasphemy, the answer is "yes" that the Romans would tolerate such actions by Paul.
