How to play sounds through AirPlay
Sorry but what you want to do is not possible.

iTunes can use AirPlay for music because it has been created to support AirPlay.  Game developers are unlikely to add support for AirPlay just for audio.  If they do support AirPlay, it would probably be to do screen mirroring or dual screen to an AppleTV.

Also, only one app at a time can use the AirPlay stream.  So even if there was a game that supported just audio to AirPlay speakers, you would not be able to have iTunes streaming to the AirPlay speakers at the same time.

This is documented by Apple for developers: https://developer.apple.com/design/human-interface-guidelines/airplay/overview/media-playback/


  Stream only expected content. Disable streaming of content like background loops and short video experiences that only make sense within the context of your app itself. 


The breakdown between game sounds that play locally and soundtracks that are presented through AirPlay is here: https://developer.apple.com/library/archive/documentation/AudioVideo/Conceptual/AirPlayGuide/EnrichYourAppforAirPlay/EnrichYourAppforAirPlay.html


  Send Audio Where It Belongs
  
  Apps typically use two types of sound—app audio (such as ambient sounds, background music, and incidental noises) and system sounds (such as key clicks and alert sounds). AirPlay attempts to deliver the app audio to a remote sound system while keeping system sounds on the host, so that feedback sounds remain local to the input device.
  
  If your app uses system sound APIs for app audio, AirPlay does not redirect them to the AirPlay-enabled sound system. The result is a less-than-optimum user experience. It is important to use system sound APIs only for system sounds. For app audio, use APIs such as AVAudioPlayer.