Thank you for all the replies!
It is a Yamaha TSR-7790, and appears to support HDMI pass through, and the Firestick has a wall wart powering it. The TV is a 9 Y/O Samsung HD TV, need to check if it even has ARC. Previously I had the Firestick just plugged into one of the HDMI ports on the TV, but now want a nice clean set-up if possible, and run the audio through the A/V system.