Steam streaming is shite with wayland but sunshine (host) and moonlight (client) work great. Fussy with HDR but fine for mere mortals.
You might be interested in a BC250. It's what I did after wanting exactly the setup you describe lol. Now I just game on the couch and stream anything CPU and ram heavy.
Stretching a cable halfway across the room makes anything with a weak Bluetooth antenna horrible :( controller dropouts and stuff