Josephus records James’s death but says nothing about why he died or what he believed. Doesn’t using his martyrdom as evidence for the resurrection stretch the source?

The two sources are kept separate on purpose. Josephus confirms that James existed, was Jesus’s brother, and was executed at the high priest’s instigation in AD 62; the specifically Christian reason for his death comes from Hegesippus, a later and less reliable source. What Josephus establishes on his own is enough: a brother who had not followed Jesus during his ministry became the leader of the Jerusalem church and was killed by the same religious establishment that had condemned Jesus. Something changed his mind, and the earliest tradition says what it was.