Google Books has served up a tidbit of information. On page 51 of the Smithsonian's Report of the Secretary and the Financial Report of the Executive Committee of the Board of Regents, we learn:
"Associate curator Edwin A. Battison, assisted by summer intern Bruce H. White,completed the first draft of a translation of Jacques Besson's ..."
And that's it. We don't even have a year! There is one clue. In a 1966 article on Besson, Battison's bio reads:
"Mr. Battison, Curator of Light Machinery in the Division of Mechanical and Civil Engineering of the Smithsonian Institution, is enraged in translating and editing the 1578 edition of Besson's Theatrum Instrumentorum et Machinarum."
The 1966 volume may be a good to place. As for finding more details, consider this personal communication from Donald Brashear:
"Thanks for your email. I can now only vaguely recall the English translation project being mentioned when we first started the Digital Besson project. I seem to recollect hearing that someone had been working on an English translation but that it did not get very far. No one seemed to know if there was any partial material remaining, and there was no follow-up.
"I'll try to do some more digging here to be sure, but I think it's pretty likely that Battison's project must be the same one and that it never was completed. A search in the Smithsonian Archives' curatorial correspondence might turn up some clues if no one here remembers."
One place to start is in the archives. The translation may be lurking somewhere in the 10 cubic feet of exhibition records stored at SIA Acc. 95-116. It also contains some of the correspondance of Eugene Ferguson. The curatorial correspondence at SIA RS00536 may also be helpful since it contains copies of manuscripts... like the completed Besson translation.