Well, the Amazon Fire stick runs off of Android, right? Android actually has a Bluetooth subroutine that asks paired Bluetooth headphones what their latency is, and if the headphone manufacturer knew about that feature and supported it, the headphone will share its latency spec and Android will delay the video to match. This is also true of iOS (and the Apple TV).
However, that doesn’t help someone who uses a game console for games or apps (like Netflix, Hulu, etc), is watching content from physical media (like BluRay, DVD), or another source (like cable TV). For that, a more universal transmitter would help, but it should have optical (preferred for SQ and latency) and an analog 3.5mm input (“aux”), support aptX LL and higher bitrate codecs (AAC, aptX HD, LDAC, LHDC, etc) than just the Bluetooth standard SBC (which is the baseline, but the old standard only sounds about as good as a medium bitrate MP3 file and has about a quarter-second latency or more), and it’s really nice if it’s a class 1 Bluetooth device with extended range.