If you have the gear, go ahead and give it a shot and see what happens!
However, I'd suggest returning the Fire and getting a Roku for 2 reasons.
The Roku has both toslink and HDMI output so you can freebase 4k to your TV and send the audio to your AVR.
When you do a search on the Roku, it will show you search results for all channels (netflix, hulu, amazon, yada-yada), if you do it on the Fire, it will return Amazon only. Amazon doesn't want you to know that you can watch a movie free on Netflix, or a dollar less on Vudu when you can pay them $$ for it). At least that is how it was a year or two ago when I bought my Roku 3.