Did James brother of Jesus, believe that Jesus is God? if so, why were some Jews in the city offended by his death according to Josephus?

It's not entirely clear, but we could surmise that like the accusations against Jesus, they were minor and did not deserve death.

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,

Breakers of the law is a wide accusation. As James wrote, this could involve one minor break and therefore be guilty of the whole breakage.

For James, if he believed Jesus was the Christ, Son of the Living God, then he wouldn't portray himself in the same way, being worshipped. So, it had to be minor.

some of them went also to meet Albinus, as he was upon his journey from Alexandria, and informed him that it was not lawful for Ananus to assemble a sanhedrim without his consent.

In fact, they accused Ananus of an unlawful assembly, rather than a matter of Mosaic law.

Lastly, by this time there were thousands of Jews who believed Jesus was the Christ. James was a leader in the Jerusalem Church.

But other of the apostles saw I none, save James the Lord's brother. Gal 1:19

So, most likely, this was a power, political struggle against two opposed factions.

The Jews were offended because of the unlawful manner in which Ananus formed the Sanhedrin of Judges, not because of how James broke the law.

What law did James break? It's not clear from the text. However, by consulting other texts (quotation of Hegesippus by Eusebius) it is plausible that it's because he, as the leader of the Jerusalem church, refused to help putting down Christian beliefs, which must have included affirming something about Jesus (sitting at the right hand of God) which must have aroused the charge of blasphemy which is punishable by death of stoning. (source: Wikipedia article section James, Brother of Jesus -- Death)

I think the text itself strongly supports the reading above:

1. The newly appointed high priest Ananus took advantage of a vacuum of power, while the newly appointed procurator Albinus was still on the road, to quickly assemble a Sanhedrin of Judges to condemn James to death by stoning.
2. James was condemned as "breakers of the law" for reason that is unclear (see previous paragraph).
3. But some who witnessed this stoning had the opinion that "it was not lawful for Ananus to assemble a Sanhedrin without [Albinus's] consent." They appealed to Albinus and King Agrippa.
4. Note that the passage did not mention that the appeal was because the judgment against James was wrong (i.e. because he convened the Sanhedrin to convict an innocent man) but simply "they disliked what was done" which is elaborated later in the passage as the unlawfulness of the proceeding. If the sentence was unjust according to the appealer, it would have been mentioned.
5. King Agrippa agreed with them and removed Ananus from office.

Based on the above reading, your statement "they saw nothing against the Law to be committed by James, to label him as a breaker of the law" is not explicitly supported by the passage, and consequently cannot be used to support the theory that elevated views of Jesus wasn't yet formed by James's death (around 62 AD).

A fuller Christian narrative about James's death from Christian perspective, which is also consistent with the above reading, can be read in this 2019 blog article The Unsettling Truth of James’ Martyred Death. Another Christian narrative is at The Death of James the Just, Brother of Jesus Christ. Another article by Sean McDowell Did James, The Brother of Jesus, die as a martyr? quoted Prof. Darrell Bock about the law that James broke:

What Law was it James broke, given his reputation within Christian circles as a Jewish Christian leader who was careful about keeping the Law? It would seem likely that the Law had to relate to his Christological allegiances and a charge of blasphemy. This would fit the fact that he was stoned, which was the penalty for such a crime, and parallels how Stephen was handled as well.

A Jewish narrative at The Stoning of James attributed the hostility to James's zealous concern for the poor and the oppressed as well as toward the wealthy, pointing to possible interpretation of James's being innocently condemned. As you pointed out in the comment, there IS something that Josephus said (according to Eusebius) which needs some reconciliation with the reading above, and which unfortunately is not included in the 2 Christian narratives mentioned above:

Josephus, at least, has not hesitated to testify this in his writings, where he says, «These things happened to the Jews to avenge James the Just, who was a brother of Jesus, that is called the Christ. For the Jews slew him, although he was a most just man.»"

We may not able to resolve whether the Christian or the Jewish narrative gives the true reason for James's stoning. I'll update the answer once I research more into this.