As you know, Lexicon's are not meant for the study of one particular author or time period, unless they were assembled specifically for that purpose.
There are two really helpful works made for Josephus, though they are not very accessible at all (if you are by a research library, they might have a copy):
Thackeray, Henry St. John, and Ralph Marcus. A Lexicon to Josephus
. (only Vol. 1 exists, unfortunately).
Rengstorf's A Complete Concordance to Flavius Josephus
. (4 volumes, covers all of Josephus). This one is not properly a lexicon, but it does provide a range of glosses for all the words in Josephus.
When I was doing a lot of reading of Josephus I found LSJ to be adequate for most words. Usually you don't have to go beyond the main senses of the words, provided you pay close attention to the normal issues like different meanings of verbs in different voices or transitivity. If you're going to be spending much time in Josephus, it is worth getting an idea of what the author abbreviations are in LSJ for the people writing around his time. With LSJ I have usually found that if there is no sense listed for a word that is specifically noted as a later meaning than the main one, it is usually adequate (though often not ideal) for any given author following the main period. BDAG is often helpful for words in Josephus, when it covers them. Similar for any LXX lexicons (if you have one). I occasionally found helpful glosses in Lampe's A Patristic Greek Lexison
(there is a pdf version of unclear legality online) and Thackeray and Sophocles' Greek Lexicon of the Roman and Byzantine Periods from B.C. 146 to A.D. 1100
as well (free fully legal pdf version online).
The process I found most helpful was to read the text alongside the Loeb translation (if you have access to it) and Michel and Bauernfeind's German translation (Flavius Josephus de Bello Judaico: der Jüdische Krieg) which comes in a 4 volume set (I got one really cheap from abebooks), if you read German. I found the German to be more transparent to the structure of the Greek than the Loeb translation, most times. The online PACE translations https://pace.webhosting.rug.nl/york/york/texts.htm
are really helpful, where they exist. There are apparently many good French translations as well, but I don't really read French. Apparently, Thackeray's English translation is often simply a translation of a certain French translation that he relied heavily upon.
Considering how much work has been done towards a Josephus lexicon in the past, it is sad that one does not exist yet. Adolf Schlatter (a famous German New Testament bible scholar from the early 1900's) had made a personal lexicon of much of Josephus, and these notes eventually made it to Rengstorf, I believe, but are lost or inaccessible (not to mention illegible: Schlatter's handwriting was terrible!). Henry St. John Thackeray did the same thing, again handwritten, and some other scholar got all the pages and they are lost to the scholarly world.