Not sure about the native app on the Firestick (don't own one and don't subscribe to Amazon Music,so couldn't test/confirm), but no device will allow you to 'cast' audio from your phone to your TV with higher bitrates. The only way this would be possible is if the streaming is taking place in the device itself and, of course, the music service supports the higher quality audio on said device.
From Amazon's help page:
You can listen to HD and Ultra HD music on Amazon Music on the following devices:
- Echo devices
- Sonos speakers
- Fire TV
- iOS devices running iOS 11 or later
- Android devices running Android Lollipop or later
- Desktop
- PC
- Mac
Now, it doesn't specifically state 'Firestick' devices, but I'd assume they'd count as 'Fire TV' ones (but again, don't own one to confirm).