Don't know the answer, but you've got to also consider in addition to what FlashGot may give away, what might Gigaget, or Orbit or ... download managers give away?
I would think
the safest route would be to forgo FlashGot & only use the browser built-in downloader.
I've got a simple question.
When it comes to anonymity nothing is simple
Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:22.214.171.124) Gecko/20110420 SeaMonkey/2.0.14 Pinball NoScript FlashGot AdblockPlus
Mozilla/5.0 (Windows NT 6.1; WOW64; rv:16.0) Gecko/16.0 Firefox/16.0 SeaMonkey/2.13a2