Because of how OSX makes the connections to AVB devices selected as local Core Audio devices, recording over AVB is not supported when using Stagebox mode with your StudioLive Series III Console and a StudioLive Series III Rack mixer.
With any network audio technology, you can have one talker go to many listeners, but a listener can only listen to one talker.
When you select an AVB device in Audio MIDI Setup Network Device Browser, OSX will actually force a connection of its own AVB streams to every talker and every listener stream of that AVB device, because it needs to receive from every talker stream of the device as well as send to every listener stream of the device.
Because AVB stagebox mode is a bi-directional relationship between the two mixers, they also need to connect all of their talker and listener streams.
As you can now see, because you can't have the listener streams on the mixer listening to the talker streams of both the other mixer and the Mac, it's not possible to have both stagebox mode in use and have the Mac connected to either mixer as a Core Audio network device.
In fact, even simply connecting a Mac to your AVB network with either StudioLive mixer enabled in Audio MIDI Setup as an AVB Audio Device will interfere with the AVB routing necessary for Stagebox mode functionality, and in effect, break stagebox mode.
So, if you are using a Series III Console with a Series III Rack mixer as a stagebox, if you have any Mac computer connected via ethernet to the same network that the Audio Network port of the two mixers are connected to, you must make absolutely certain that the Mac does not have either mixer checked off in Audio MIDI Setup Network Device Browser to be used as a local Core Audio device.
When in doubt, the safest bet is to just entirely avoid connecting the Mac's ethernet connection to the same AVB switch/network as the mixers' Audio Network ports.