Voila! I unloaded the Black PLA from the second print core (a BB. 0.4 core) and indeed, all the factory profiles show up. Great to have a solution on this, but would prefer (because I'm a delicate little end user) to have the app inform me of this in some way, or if selecting a factory profile, have it confirm that it will NOT use any print cores with unsupported materials in them and list the problem it sees.
But for now, I AM HAPPY! 🙂