Does the TV do ARC? Audio Return Channel via HDMI to the amp, basically two way between amp and TV. Often only one socket on the TV will do it, and you may have to enable the amp to do ARC.
With that set up, you can hook the Amazon box straight to the TV with HDMI and the HDMI connection to amp will do return audio when you play things via the TV. Should be max quality audio, unlike optical audio which can be limited. No need for splitters and video is 4k direct into TV, whereas the amp may not cope passing that through.
All said, I suspect you won’t be fussed after the initial novelty of the small amount of 4k content has worn off 😉