Found the problem.
I had MIC+LIN set ON for USB, but that is per-mode, and it was OFF for DATA.
So turning on MIC+LIN ON for DATA, PTT over CAT now works as it should with fldigi and WSJT-X.
I guess with MIC+LIN OFF, activating the front-panel-connected foot switch directed the K3S to take audio from the front panel, which it otherwise wasn't doing? Got to say, I don't understand some things about this setup.
For example, trying to fix this, somewhere I made some change where now CW doesn't work! But, onward.
73, Paul K6PO