I recently bought a Fire Stick Lite to use with my projector. It plugs into the HDMI port of the projector and then I connect my projector’s Bluetooth to my Alexa (Echo Flex, plugged into speakers and subwoofer).
However, this would lead to the audio becoming more and more delayed the longer I watched a movie. I tried the audio delay sync setting (AV Sync Tuning) with no success.
My solution for this was to connect the Bluetooth directly to the Fire Stick instead of the projector (or your TV). This completely removed any audio delay and the issue is now (thankfully) resolved.
In the settings menu of the Fire Stick, go to ‘Controllers and Bluetooth Devices’ and then select ‘Other Bluetooth Devices’. You can then pair your Bluetooth speaker directly with the Fire Stick. If you are pairing an Echo Dot, you can open the Echo Dot in the Alexa app on your phone and set up pairing that way.
It is worth noting that it recommends using a 5GHz wifi network for the Bluetooth device, but it is working fine on my 2.4GHz network.