I noticed that Eusebius transcribes the entirety of Josephus' Testimonium Flavianum in 324 AD, and that the First Council of Nicea occurs in 325 AD. Does any of Josephus' work feature throughout the contemporary arguments or literature?
The most important early Church documents that relate to the history of the First Council of Nicaea written by contemporaries of the period are, I think, the Acts and Canons of the Council; the Ecclesiastical Histories of Eusebius (263-339); and the writings of Athanasius (296-373).
Most of the Acts of the Council have not survived (the Nicene Creed is an example of a portion that did). There is no mention of Josephus in the portion of the Acts that we have, nor in any of the Canons.
You have already pointed out that Josephus was mentioned in Eusebius. References to him occur in least once in Books I, II, III, IV, IV, VI, VII, but as far as I can tell, nothing that is written is relevant to the Arian controversy, which was the subject of the Council.
Most, if not all, of Athanasius' works related to the Arian controversy, and hence the Council, can be found in the CCEL Nicene and Post-Nicene Fathers collection, but I cannot find a single reference to Josephus by Athanasius in any of the documents available.
We might consider also the Church Histories of Socrates of Constantinople (380-439), Sozomen (400-450), and Theodoret (393-458). All three of these wrote on the Council of Nicaea. Socrates mentiones Josephus in Book V of his history, but on an unrelated matter (the celebration of Easter). Sozomen mentions him twice, but again in completely unrelated contexts. Theodoret does not seem to mention him at all.