I can tell you - use cables when ever you can. Easy and stable connection. And if you have the right gear, you will have no noise at all (thats how we do it as DJs etc - and we have more issues with wireless stuff than with cables).
Its still a mess because how ever you do the audio routing - you will still have 2 sources (desktop audio with the game sound & background music).
Its not AirPlay, its the system with 2 different audio sources. If you want to run your music on a different output (same PC) or even external source connected to your A/V-receiver.
The best way is to run the desktop output & music on the same output on your PC. That will work without additional software and this is how 95% people handle the audio while streaming.
So you have to grab the source and mix it with your mic input or your desktop input and then select the combined input/output in OBS.
Right now OBS can handle only 2 audio sources, you have 3 - music, desktop audio & your mic.
I dont know the plugin, i dont use the apple stuff much. Only for DJing & music production, but no wireless connection is used there.
Cable will do the job just fine and you dont have to mess around with even more software, but its your system - try what works best for you. Wireless stuff can also add some latency - dont forget this.