The Chronicle of Eusebius ended with the Vicennalia of Constantine; that is clear from Jerome’s translation/expansion of it in Latin. From Michael the Syrian we learn that Jacob of Edessa commenced his Syriac continuation at the same point.
Looking at Michael’s text, it is clear that the Testimonium Flavianum quoted in it comes from the same material as the mention of Phlegon; that is, presumably from Eusebius Chronicle via some Syriac translation.
I think that we can presume that, like other works of Eusebius, the Chronicle was translated into Syriac early. Indeed that Jacob in the 7th century uses the same end point as Jerome in the fourth says that he was working with a translation made fairly early, as he then has to supplement it with material from Socrates and Theodoret! A later translation would probably have been from a revised edition of the Chronicle such as that of Annanias or Panodorus, or betray the signs of reediting that the Armenian translation does (itself a 5th century product). Both books must have been translated, judging from the presence of material from book 1 at the start of Michael the Syrian.
But the Testimonium never formed part of the Chronicle, so must be an addition, and probably after the text had been translated into Syriac. It seems unlikely that Josephus Antiquities was translated. But we do know that Eusebius Church History was, since it is extant in that language. What, then, is the version of the TF in that text?
Most people are familiar with the TF in Michael because it was published by Shlomo Pines when he published the version in Agapius. But although a printed edition of the Syriac Eusebius HE exists, no translation exists. The same applies to the Armenian text of the HE.
One interesting feature of Michael’s quotation is that it agrees with Jerome’s Latin version “He was thought to be the Christ”. If this did not come from Josephus, does it mean that the HE in Eusebius originally read thus?