@mitu
Ok, I solved the issue.
The pad has a switch on its back to commute between different modes (Windows, Mac and so on).
It must be set to 'X' in order to be used with Raspberry; I found this suggestion on a Reddit post, since it wasn't mentioned in the walkthroughs I read on the web (or at least I missed it)...
By the way, I paired the controller again and tested it successfully with several games!

@YFZdude I see. Thanks for the info and the tip. But the issue I have, and the reason I wanted for every button to work on d-input, is the fact that both controllers have the same name in switch mode, which is a problem with their wildly different layouts.

Solved

@mitu This seems to have solved the issue, thanks!

Just to close the loop on this. I was unable to try the above solution as the controller had paired, but no security settings or configuration was allowed. When I tried to unpair the controller I couldn't. I assume I had messed up the settings trying fixes for this. I uninstalled retropie and reinstalled and it worked fine. Thanks for the help.

@x_wav This means the button is not registering, so it can't be used.

@mitu IT WORKED!!!!!

I needed to install xpadneo. Thank you so very very much - I spent so long trying all the different combinations!!