This would require a virtual loop device so that the system audio can be used as an input stream for the LMS (e.g. with the wave input plug-in). Unfortunately, we don’t offer something like this. I would recommend looking for ALSA Audio Loopback on the Internet and maybe developing a solution yourself.
In order to play the system audio on your Max2Play device it should be enough to deactivate the integrated audio of the Pi in the Raspberry Settings and to stop Squeezelite and Shairport, or to set the „-C“ parameter so that they are automatically stopped when not in use.