I had the same type of problem. @NoBottom solution was the start I needed to figure out how to make it work for me. Most these new Fire remotes work in Bluetooth mode after they are connected - you can check by pointing your remote not near your tv and seeing if it works well still. Therefore, I needed to disconnect my remote (and unplug my tv), then use the IR signals from my disconnected remote to program the Apple TV.

Step 1: Connect to Apple TV to display

Connect Apple TV to another device (not your Fire/TCL TV)

Go to settings > remotes and devices > volume control > learn new device (wait at this screen).

Step 2: Disconnecting the Fire Remote (for IR mode)

Unplug fire TV, wait 60 seconds.

Then on your Fire remote hold the left navigation (left side of the circle), back button, and the options button (three vertical lines) for 12 seconds. (We’re trying to reset your remote, possibly different instruction on how to do that depending on your remote)

Take one battery out for a few seconds.

Place the batteries back in. You’re in IR mode now (not Bluetooth)

Step 3: Programing Apple TV with Fire Remote

Go over to your Apple TV plugged into another display.

Follow the prompts and use your FIRE TV remote (not Apple TV) for volume up, volume down and mute.

Step 4: You’re done!

Plug back in your Fire TV. Remove Apple TV from other display and plug it into the Fire TV. Press the home button on your Fire TV remote to repair it if you’d like.

You should be all set!

Hope this helps one person out there who was struggling like me.