In case anyone else ever finds themselves in this particularly stupid situation: Reinstall Pick Master 3 on the PC. That's it. A simple reinstall fixed the issues because a particular runtime app did not install properly the first time around and I ended up wasting weeks and everyone's time on a dumbass problem.
It's only mentioned in ONE place in the manual and we had to use an external app to monitor PC - Robot communication and discover what we were missing.