To conclude this thread: It turned out that our user had been registered as a participant in a BIM360 (C4R?) trial about two years ago. That trial period is well and truly expired. The system still allowed the user to have visibility into the BIM 360 design area but no permissions to edit files. Autodesk kindly extended the user's trial to the end of the current trial period and he was immediately able to access and edit files. All good now.

Interesting to note for future reference:
* Our user has no recollection of doing a C4R trial two years ago. (His focus is on interior design, not on what he may or may not have done years ago.)
* We could not see any indication or record on his user account settings that this trial had taken place.
* The Autodesk technicians helping us did not ask this question and took a long time to figure it out.
* This whole problem only happened because we were participating in Autodesk's generous extended trial. If we had purchased the licenses, all would have been fine.

We know all this now, but we did not know it then. What a waste of time for all those concerned. But at least we now have more knowledge.