I would love to be wrong about this, but it appears as though Amazon’s 2nd generation Fire TV Game Controller and 2nd generation Fire TV Voice Remote are not compatible with Element’s Fire TV Edition televisions. Nothing I tried could get them to pair with the TV. It’s not too big of an issue that the voice remote doesn’t work, as long as you don’t lose the one that comes with the TV, since Amazon does not yet sell replacements. The bigger issue is the Fire TV Game Controller, since that is the only official game controller that Amazon makes anymore.
The reason these peripherals don’t par with the TV is likely because they are WiFi-direct based remotes and game controllers. The Alexa voice remote that comes with the TV uses Bluetooth to communicate, just like the 1st generation Fire TV peripherals.
Speaking of the 1st generation peripherals, I was able to successfully pair all of them with the Fire TV Edition television. That includes the Fire Game Controller, old Fire TV Voice Remote, and the Fire TV Stick Non-Voice Remote. The only issue I found is that microphone input for Alexa commands and voice searches through the 1st-gen voice remote does not work. When you press the microphone button, the TV responds as it should, but your voice is not transmitted to the TV, so all Alexa commands and searches fail.
Not having a readily purchasable official game controller that works with the Fire TV Edition televisions is a shame, especially since I think the 2nd-gen game controller is a great product. Hopefully this is just a launch bug or an oversight that will be fixed with a future software update. Amazon did add support for the new WiFi-direct peripherals to the older 1st generation Fire TV and Fire TV Stick when the new peripherals were released with the 2nd-gen Fire TV, so it seems like it should be possible to do the same for the Fire TV Edition televisions. In the meantime, at least the Fire TV Edition televisions are compatible with plenty of high-quality 3rd-party Bluetooth game controllers, like the PS4 DualShock 4 controller.