Let's also bear in mind that the Pharisees (who were in charge of the Sanhedrin during this period according to Josephus) tended to not execute people, so one might assume this was the case regarding non-priests being in the Temple (as per Sanhedrin above).
Ant. 13.10.6:
Now there was one Jonathan, a very great friend of Hyrcanus's, but of the sect of the Sadducees, whose notions are quite contrary to those of the Pharisees. He told Hyrcanus that Eleazar had cast such a reproach upon him, according to the common sentiments of all the Pharisees, and that this would be made manifest if he would but ask them the question, What punishment they thought this man deserved? for that he might depend upon it, that the reproach was not laid on him with their approbation, if they were for punishing him as his crime deserved. So the Pharisees made answer, that he deserved stripes and bonds, but that it did not seem right to punish reproaches with death. And indeed the Pharisees, even upon other occasions, are not apt to be severe in punishments.
But both Josephus and Hegesippus present James' death as occurring during a near-anarchic state of affairs (with the former being instigated by a Sadducee), which is also in keeping with Pes. 57a above.
Ant. 20.8.8 (right before the James passage):
And now arose a sedition between the high priests and the principal men of the multitude of Jerusalem; each of which got them a company of the boldest sort of men, and of those that loved innovations about them, and became leaders to them; and when they struggled together, they did it by casting reproachful words against one another, and by throwing stones also. And there was nobody to reprove them; but these disorders were done after a licentious manner in the city, as if it had no government over it. And such was the impudence and boldness that had seized on the high priests, that they had the hardiness to send their servants into the threshing-floors, to take away those tithes that were due to the priests, insomuch that it so fell out that the poorest sort of the priests died for want. To this degree did the violence of the seditious prevail over all right and justice.
Ant. 20.9.2 and 4 (right after the James passage):
But as for the high priest, Ananias [the father of the priest who killed James], he ... had servants who were very wicked, who joined themselves to the boldest sort of the people, and went to the thrashing-floors, and took away the tithes that belonged to the priests by violence, and did not refrain from beating such as would not give these tithes to them. So the other high priests acted in the like manner, as did those his servants, without any one being able to prohibit them; so that [some of the] priests, that of old were wont to be supported with those tithes, died for want of food.
Costobarus also, and Saulus [who I think could be Paul], did themselves get together a multitude of wicked wretches, and this because they were of the royal family; and so they obtained favor among them, because of their kindred to Agrippa; but still they used violence with the people, and were very ready to plunder those that were weaker than themselves. And from that time it principally came to pass that our city was greatly disordered, and that all things grew worse and worse among us.
EH 2.23.10:
... there was a commotion among the Jews and Scribes and Pharisees, who said that there was danger that the whole people would be looking for Jesus as the Christ.
EH 2.23.16-18:
And they cried out, saying, 'Oh! Oh! The just man is also in error.' And they fulfilled the Scripture written in Isaiah, 'Let us take away the just man, because he is troublesome to us: therefore they shall eat the fruit of their doings.'
So they went up and threw down the just man, and said to each other, 'Let us stone James the Just' ... And one of them, who was a fuller, took the club with which he beat out clothes and struck the just man on the head. And thus he suffered martyrdom.
Pes. 57a:
Woe is me due to the High Priests of the house of Baitos, woe is me due to their clubs. Woe is me due to the High Priests of the house of Ḥanin [the family that killed James]; woe is me due to their whispers and the rumors they spread. Woe is me due to the High Priests of the house of Katros; woe is me due to their pens that they use to write lies. Woe is me due to the servants of the High Priests of the house of Yishmael ben Piakhi; woe is me due to their fists. The power of these households stemmed from the fact that the fathers were High Priests, and their sons were the Temple treasurers, and their sons-in-law were Temple overseers. And their servants strike the people with clubs, and otherwise act inappropriately.
So it was a crazy situation and all the rules did not necessarily apply. Maybe some (more lenient) Pharisees were cool with a Nazirite being in the "holy place" and other people weren't given these circumstances, and the people who did not like James (like the people who killed lower priests in Josephus) used this opportunity to kill him. In other words, I don't think priestly or legal procedures were necessarily being carried out by the book during this time, and it was also more of an ad hoc assembly that "came together" to take out James. And maybe this has something to do with why Ananus (who was a Sadducee) "had formed an accusation against them [James and others] as breakers of the law."
Ant. 20.9.1:
But this younger Ananus, who, as we have told you already, took the high priesthood, was a bold man in his temper, and very insolent; he was also of the sect of the Sadducees, who are very rigid in judging offenders, above all the rest of the Jews, as we have already observed; when, therefore, Ananus was of this disposition, he thought he had now a proper opportunity [to exercise his authority]. Festus was now dead, and Albinus was but upon the road; so he assembled the sanhedrim of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others, [or, some of his companions]; and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned.
EH 2.23.10:
... there was a commotion among the Jews and scribes and Pharisees ... Coming therefore in a body to James ...
Cf. Mk. 14:53 regarding the trial of Jesus, which uses the same word Hegesippus uses above (and likewise mentions three groups of people, with "the elders" presumably being Pharisees, as per Mk. 7:5):
They took Jesus to the high priest, and all the chief priests, the elders and the scribes came together.
Ant. 20.9.1:
... but as for those who seemed the most equitable of the citizens, and such as were the most uneasy at the breach of the laws, they disliked what was done.
Cf. EH 2.23.18:
And while they were thus stoning him one of the priests of the sons of Rechab, the son of the Rechabites, who are mentioned by Jeremiah the prophet, cried out, saying, 'Stop. What are you doing? The just one prays for you.'