If you are hellbent on using Josephus to prove Jesus's existence, the smart play is to sidestep the Testimonium Flaviam entirely and instead invoke the reference in Book 20, Chapter 9:
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: