How do I update the remote client remotely?
In case it helps anyone else ...
I first tried on a local machine so I could see what was involved. The new update package first gave an option "Another Version of Remote Desktop is running. Close Remote Desktop before install?". With options of Yes, No and Cancel. I chose No in the hope that it would let me install the two versions side by side temporarily but that just got me another window with "Close Remote Desktop to continue" with Retry & Cancel options.
Selecting Yes closed my current remote session and then presented a normal installer, which required being able to click "next" a few times using the local mouse. Once completed, the remote client restarted fine and I was able to reconnect.
So all it would have needed was a "run automatically/accept all" option on the initial message screen to let me run it completely remotely. That wouldn't cater for all potential problems on the install, but would at least give a chance of successfully updating.
As it is, the only solution I can come up with is to install a different (well-known name) remote access software and use that to remotely run the installer. Or maybe I'll just stick with that in future.