When I pair my bluetooth headphones to my Fire TV there is no lag. When I run Kodi there is a noticeable 1-2second lag with audio. This includes system sounds as well as playback.
I have a 1st and 2nd generation Fire TV box (no sticks) that I am able to provide feedback with if necessary.
There are other threads that are old about this issue but I have never found a fix anywhere. And no, changing the audio offset within media playback is not a solution.