Perhaps I'm just paranoid, but I'm probably not going to run through the Korg.
I'll look for a way to split the signal out of the wireless.
I would agree with you and specifically didn't want to run through my Korg rack tuner. Why take the chance of degrading your core tone?
This is the reason I programmed a silent rig with direct out set to gtr out. The Kemper taps the signal for the tuner and is constantly working, even when the silent rig is selected. As long as you have a way to change rig by hand or midi, it works best for me.
I'm a bit nervous about clicking over to the tuner with the chicken switch in case I go one too far in a gig situation! It would be a minute down while it reboots.
My bass player has taken to a Snark clip-on tuner that seems to work really well with ambient noise around.