After speaking with an Autodesk developer we now know that our issue is related to past eTransmit use. Revit mistakenly retained a flag it uses to mark that file as such.
If we can successfully use Synchronize with Central and we get that message when we use Save then the correct response to the warning dialog is the top option: “Save this model as a central model in its current location – Revit will remove this message and allow users to create local copies of the model.”
Remember, the message is accurate for files that have been created via eTransmit too. In our situation the files were working fine on BIM 360 but they retained the flag that should have been removed when they were added to the cloud.