Prime app on Apple TV - failed to generate code to register device
I recently had to delete and re-install the Prime Video app on my Apple TV 4K (the app had crashed and would only display a black screen) and haven't been able to log back in since. Every time I navigate to Sign In from within the Prime app (to register the device on my account), I get an error message saying "Failed to generate code", and the QR code is greyed out. I've used the "get a new code" button many times but I keep getting the same "failed to generate code" error.
I spoke to both Amazon and Apple support and have confirmed that it's an issue on the Apple side, but Apple Support could only suggest taking my hardware in to the store, even though everything else on my Apple TV works fine.
Has anyone encountered this issue and found a way around it?
Apple TV 4K, tvOS 16