Android beta builds have their own announcement threads (linked below).
Use the Amazon Fire OS Devices builds for Amazon devices. The Amazon AppStore build is for non-Amazon devices.
To sideload on FireTV devices:
Uninstall current app
Restart FireTV device
Use Downloader app (in Amazon store) to download and install beta release
Launch app, authenticate, etc
Note sideloaded apps do not auto-update like apps from the app store. You have to manually install each release. To go back to the production releases, simply uninstall the beta release, restart the device, and install the version in the app store.
FYI, not sure restart step is actually necessary. Just me being overly cautious about clearing out leftover stuff after uninstalls. I do the same on PCs, etc.