Just add Flirc for Fire TV as a component on your Harmony remote and. I know there are IR/RF extenders/converters out there, but I have no idea whether the Harmony 650 will work with them. Out of the box support for Logitech Harmony remote controls with the Harmony FireTV Device. For example, I have several Roku boxes (XDS, etc.) that use IR and therefore work just fine with the Harmony 650, but I recently purchased (and subsequently returned) a Roku stick that uses RF and therefore does not work with the 650. If your devices use RF instead of IR, the 650 won't work with them. Launch the Harmony software and add a new device as follows: Device: Media Center PC Manufacturer: Flirc Model: FireTV / Kodi / Plex. The easiest way to tell if your device uses IR is if the remote requires line-of-sight access to the device (can't have the device hidden in a cabinet). Sources: HRHR Prime with Charter CableCard. Most consumer media devices use IR remotes. Buy Fuzzy a beer (Fuzzy likes beer) unRAID Server: i7-6700, 32GB RAM, Dual 128GB SSD cache and 13TB pool, with SageTVv9, openDCT, Logitech Media Server and Plex Media Server each in Dockers. The main catch here is that the 650 is an IR (infrared) remote, not an RF (radio frequency) remote. In worst case, if you have the original remotes for your devices, you can "teach" the 650 to work with the buttons on those remotes.
Logitech appears to regularly update the device database that's used when you configure your Harmony remote. you are going to need a USB extender cable but connect that the that back of your Amazon TV. Sync up your Harmony and once it says you are done, disconnect it from USB. MyHarmony software to Add A Device -> Manufacturer Flirc, Device Model Number FireTV.
Setup your Harmony 650 for a new device to be. How to launch apps on Fire TV with Harmony Remote - YouTube.
Dunno for sure, but I'm guessing it will. Connect Harmony 650 via USB to your computer and start up MyHarmony software.