As ScruffR pointed out, the Core does not have a separate image for Device OS. Your user firmware binary contains both your custom firmware and the version of Device OS. So saving your firmware binary is sufficient for restoring Device OS and your firmware (on the Core only).
The second problem is that you can’t flash every conceiable version of Device OS from the Web IDE. The error message is misleading on the Core - it’s not the firmware on the device that’s the problem, it’s that the version you selected is not supported on the cloud compiler.
Version 0.5.4 is very close to 0.5.1 and that is supported by the cloud compiler.
Thanks for the feedback. I realise at this point my particular situation is now solved - since the application firmware contains the deviceOS in the case of the Core, and 0.5.4 does work, and that firmware version can be compiled and downloaded from the webIDE, then together combined I’ve got the firmware i’m looking for.
… still doesn’t address the issue in the topic title, of course. But I suppose rather than beat a dead horse I’m willing to close this thread.