This is a matter of opinion. You have the ability to several completely independent setups.
Use the Studio 192 for in-home recording with the central station as your monitor controller and take the StudioLive with you on gigs.
Use Capture with the StudioLive III to record your live audio, then load the session into Studio One and use Studio 192 as your audio I/O.
Other options to consider:
Record into and mix on the console and just use Studio One as a tape machine
or
Record into and mix in Studio One and use the StudioLive III as a controller running in DAW Mode
or
Record into Studio 192 and use the StudioLive III as a controller only (without the audio)
or
Record and Mix with Studio 192 and Studio One and do all your mixing in Studio One as the mixer for Studio 192 is controlled by Studio One with seamless integration.
in either situation you can can connect both the StudioLive III and Studio 192 main outs into the central station and switch between the two using the master knob on the Central Station to control levels to your monitors.
another option.
Central Station functions are built into the Studio 192. If you don't use those features in Studio 192, You can use the output of the Mixer and/or the Studio 192 into the Central Station and use that to connect and control levels to your monitors.
Ultimately the decision is up to you. You don't have to use the Studio 192 and the StudioLive III in the same setup.
Perhaps consider posting in the StudioLive III Community Forum or FB Groups and ask the same question there to get more feedback from others.