2023/11/01: BitGym v3.7.06 onwards has begun limited native support for Fire TV devices. This includes Amazon Fire TV Stick, Cube and Box devices, certain smart TV models and set-top box models that run Fire OS.
Fire TV Stick 4K Max (AFTKA)  is our baseline reference model. Devices older than this are less likely to run without issue.
Fire OS 7 or newer is currently supported.
How do I know if my device runs Fire TV/Fire OS?
You can search for your device model on this Amazon developer page. This page will also tell you the available Fire OS version for your device.
I can't find BitGym on the Amazon App Store on my Fire OS device.
If your device isn't able to find BitGym from the Amazon app store, it means it is not natively supported, and would require significant hardware or software workarounds.
In such cases you may still be able to "sideload" BitGym to your device. You can reference HowToGeek's guide on sideloading apps to your Fire TV Stick.