There may be multiple beta releases. The current beta may not be the version that becomes the public release. There is no publicly announced timeline for releases.
The public and beta releases cannot coexist. Only one copy of the Plex app can be installed. Delete the version from the Amazon app store before installing the beta.
Sideloaded apps do not auto-update. To go back to the public version, delete the beta and install from the Amazon store. To update to a new beta, delete the current and install the new version.
The Fire TV must be configured to allow apps from unknown sources (sideloading). Fire TV Settings → My FireTV → Developer Options → Apps from Unknown Sources = On.
You can use the URL in the Plex Android TV Beta Announcement.
Use the Amazon AppStore (armv7a) version. The Google Play versions will not function correctly on Amazon devices (Amazon uses a customized version of Android).
Use a URL shortener such as tinyurl.com or bitly.com to make entering the URL in Downloader (much) easier.
You might be able to use the Amazon Fire TV app to more easily enter the URL. The app acts as a remote control for the Fire TV. I’ve never tried using it to enter a URL.