Your library can be accessed on Amazon Fire TV devices by using the GSE IPTV Player App. To install the app, you'll need to sideload the application. Directions for doing that can be found here:

If you're loading apps from your Android phone (which is one of the options from the above links) then simply search GSE IPTV in the Google Play store and transfer over. If you're using the direct download method via the Downloader app, you can use the following link to download the GSE IPTV APK file (click the "Download for amazon devices" button):

Once you have the GSE IPTV app installed, add your channel PEAR library URL to the remote playlist section. After that simply launch the playlist and you're good to go. You'll need to reload the playlist every 24 hours to keep the links current.