Well, they really dropped the ball on this one. How on earth can a trillion dollar company not make a remote with a few universal buttons to control a TV and sound bar? Then they come out with this mess? Someone is losing their job over this one.
Anyway, this is remote is going to work for you or not. There's no way for you to tell before you buy it. What will happen is that once you pair the remote with your Fire TV it should put up with a "Hey, do you want to set up your TV?" prompt. If it doesn't, send it back. It looks like it controls stuff over HDMI and not IR. So if you have any sort of video switcher between your Fire TV you are out of luck. Why not just add an IR LED to this thing? Who knows.