Officially, Rome was tolerant of Judaism, and acknowledged it as a legal religion. In the first few decades after Easter (including the events in Acts referenced in the OP, in the early 40s), Rome saw Christianity as a part of Judaism—examples include Paul’s trial before Gallio, and Claudius’ expulsion of the Jews from Rome.
Paul’s trial before Gallio
When Paul was taken before Gallio, Gallio considered the dispute between Jews & Christians as questions of Jewish law.
From Acts 18:
12 And when Gallio was the deputy of Achaia, the Jews made
insurrection with one accord against Paul, and brought him to the
judgment seat,
13 Saying, This fellow persuadeth men to worship God
contrary to the law.
14 And when Paul was now about to open his mouth,
Gallio said unto the Jews, If it were a matter of wrong or wicked
lewdness, O ye Jews, reason would that I should bear with you:
15 But if it be a question of words and names, and of your law, look ye to
it; for I will be no judge of such matters.
Claudius expulsion of the Jews from Rome
Around AD 49, Claudius declared the expulsion of the Jews from Rome (see Dio Cassius, Roman History 60.6.6-7 & Paulus Orosius, Historiae adversum paganos 7.6.15-16)
This event is also recorded in Acts:
And found a certain Jew named Aquila, born in Pontus, lately come from
Italy, with his wife Priscilla; (because that Claudius had commanded
all Jews to depart from Rome:) and came unto them. (Acts 18:2)
It is clearly noted in the history of Paul’s work that Aquila was a Christian; he is referred to in Acts as a Jew, and Claudius’ expulsion of Jews from Rome applied to him as a Christian.
--
So at the time of Herod Agrippa I (died ~AD 44), in practice, the examples we have indicate that Christianity was considered a part of Judaism, and therefore a legal religion.
--
There were exceptions
The fact that Judaism, and with it Christianity, were legal religions did not mean that they were never persecuted. A noteworthy exception is the anti-Semitism of Sejanus, head of the Praetorian guard, and for a time the right-hand man of Emperor Tiberius. Philo indicates that Sejanus was deeply antagonistic to the Jews, and after Sejanus’ death (AD 31), Tiberius acknowledged the empire was going in the wrong direction and tried to reset tolerant treatment of the Jews (see Philo "On the Embassy to Gaius" ch. 24 here)
The OP referenced the placing of Roman artifacts in places considered sacred by the Jews. This never went over well. Two notable examples included the Pliate’s golden shields incident, which earned him a solid reprimand from Tiberius (see discussion by Maier here) and the notorious effort by Emperor Caligula himself to put his image in the Jewish temple (Caligula was assassinated before he could make this happen, but he caused no small stir in the attempt). See discussion on this site here.
--
We’re still talking about a time several decades before Titus destroyed Jerusalem, where setting up Roman icons in the Jewish temple would not be considered an appropriate action let alone an official responsibility. Neither would requiring Jews/Christians to make sacrifices to Roman gods.
--
Herodians
The Herodian dynasty were ethnically Idumeans, but presented themselves (with varying degrees of success) as observant Jews. The aforementioned article by Maier points out that the Herodians came out in opposition to Pilate during the golden shields incident (this was during the lifetime of Agrippa I but before his tenure as king), siding with the outraged Jewish population.
Additionally, Acts 18:3 (noted in OP) suggests Agrippa’s actions against Christian leaders were intended to please the Jews, not to persecute them.
--
Division of religions
The full separation of Judaism & Christianity into distinct religions did not happen until the Flavian era (70s-90s), and was closely tied to the destruction of Jerusalem—see discussion by Edmundson here. Thus, Roman rulers who were less-steeped in Judaism would have seen Christians as Jews, and Roman rulers, like Agrippa, who were very well-informed in matters of Judaism would be more likely to have seen Christians as apostate Jews.
Incidentally, in his Gospel & the book of Acts, Luke is careful to present the Romans as respectful and tolerant (e.g. the Gallio incident, Pilate’s desire to release Jesus). Luke was writing to an educated Greco-Roman audience, and his works presuppose an era before Rome started organized persecution of Christians. Some have even argued that the book of Acts was written as part of Paul’s defense in his trial before Nero (see here). In Luke’s writings, the persecution of Christians comes from the Jews, not the Romans.
--
Conclusion
Agrippa would not have had grounds to persecute Christians as an illegal religion in the 40s; he may well have been sympathetic to Jewish leaders who saw Christians as a perversion of Judaism, and therefore executed James to win the favor of the Jewish hierarchs.
Putting down revolts would be within Agrippa’s purview. Requiring Roman sacrifices or images in the Jewish temple would not. Demanding that Christians recant their belief was a feature of Roman rule that was still several decades away; it does not appear that anything like this was done by Roman rulers within Agrippa’s lifetime.
P.S.
Note that the passage in Revelation speaks of an event for which history has not preserved specific details. However, even those who hold to an early date for Revelation acknowledge it was written no earlier than the late 60s, after the Roman persecution of Christians had begun--the political environment was very different from the time of Agrippa's reign--presumably then if Antipas was put to death by Rome, it would have been no earlier than the Neronian persecution described in 1 Clement 5.
