Read up on the studies. My most interesting sources for were Feldman and Vermes, Got a copy of his work on Josephus work online. The only potential issue I found is the potential positive mention attributed with the term "wise man" but the fact that it contradicts with NT literature
would speak against it as interpolation. Though I will say that you may be right about 1 thing. "the term called Christ" might not be a decidedly negative mention. I might be an error to describe it negative, but I would say that Josephus would use the it in the neutral. Possibly saying that Jesus was a good man but not a Messiah figure, similar to his fairly positive mention to John the Baptist.
Here is some data by Vermes describing Josephus use of that Passage.
"Renowned Jewish scholar Geza Vermes also believes that the Testimonium originates from Josephus, albeit with Christian additions. Vermes demonstrates that the expressions 'wise man' and a 'performer of astonishing deeds' are thoroughly Josephan in style:"
"(1) The form of the description of Jesus as sophos aner and paradoxon ergon poietes, when compared with the presentation of other personalities, biblical and post-biblical, strikes me as genuinely Josephan. King Solomon is referred to as 'a wise man possessing every virtue' (andri sopho kai pasan arÍten echonti) (Ant. viii 53). The prophet Elisha was 'a man renowned for righteousness' (aner epi dikaiosune diaboetos) who performed paradoxa erga (Ant. ix 182). Daniel, in turn, is portrayed as 'a wise man and skilful in discovering things beyond man's power' (sophos aner kai deinos heurein ta anechana) (Ant. x 237). A little later he appears as 'a good and just man' (aner agathos kai dikaios) (Ant. x 246). Ezra is said to have been 'a just man who enjoyed the good opinion of the masses' (dikaios aner kai doxes apolauon agathes para to plethei) (Ant. xi 121). Among post-biblical personalities, Honi-Onias is called 'a just man and beloved of God' (dikaios aner kai theophiles) (Ant. xiv 22), and Samaias 'a just man' (dikaios aner) (Ant. xiv 172). John the Baptist is introduced as 'a good man' (agathos aner) who 'exhorted the Jews to lead righteous lives, to practice justice towards their fellows and piety towards God' (Ant. xviii 117). As for the leading Pharisee at the time of the outbreak of the first revolution, Simeon ben Gamaliel, he is presented as 'a man highly gifted with intelligence and judgment' (aner pleres suneseos kai logismou) (Vita 192) ..."
"In brief, there seems to be no stylistic or historical argument that might be marshaled against the authenticity of the two phrases in question. In fact, the clause that follows 'wise man', viz. 'if indeed one might call him a man' (eige andra auton legein chre), which is generally recognized as an interpolation, seems to support - as Paul Winter has aptly pointed out - the originality of sophos aner, an idiom which in the mind of a later Christian editor required further qualification."
"(2) In addition to appearing prima facie to be Josephan, closer analysis of sophos aner and paradoxon ergon poeites points to the improbability of their later Christian provenance. To begin with, the title 'wise man' has no New Testament roots, and in the absence of such an authoritative backing it is, I think, totally unfit to express the kind of elevated theological notion that a forger would have intended to introduce into Josephus' text. It would have been meaningless to invent a testimony that did not support the belief of the interpolator. But not only does it fail to convey the idea of the divine Christ of the church; it actually conflicts in a sense with New Testament terminology. Jesus is admittedly twice identified by Paul in I Cor. 1:24 and 30 with the abstract 'wisdom of God', but the adjective sophos as applied to men in the same chapter (1:18-31) carries a pejorative connotation. Furthermore, on the only occasion where the Gospels put this word into the mouth of Jesus, 'the wise' are unfavourably compared to 'babes' (nepioi) (Matt. 11:25; Luke 10:21). In the few instances where the term sophos is employed positively, it relates to Christian teachers, but never to Jesus himself. (Vermes, Jesus In His Jewish Context [Fortress Press Minneapolis, 2003], pp. 92-93; bold emphasis ours)"