Disclaimer, I have neither 4k tv (still 1080p like you) nor Atmos speaker setup. Unless Amazon did something weird I just don't see why Atmos would be dependent on a 4k tv as you're asking (I understand the stick itself can be limited in capability on audio or video aspects).
I understand what you're asking, but don't see why they would tie together the video and audio like you were asking. Atmos soundtrack doesn't require 4k video on discs, just the processing capability for either in your player/avr.
HDMI should be backwards compatible for 1080p, like a 4k blu ray player being able to use a non-4k tv set.
I could be wrong, hopefully someone can confirm.