The 8bitdo Ultimate 2c wireless controller has two modes: 2.4G (using a proprietary dongle) and Bluetooth (BT). Both methods of connecting the controller and playing in Steam work fine but both methods are incomplete:
When connected using 2.4G, the extra L4/R4 buttons can not be configured in Steam. The option to activate extra buttons appears in Steam but it won't work and deactivate itself again.
When connected using BT, the extra buttons can be configured just fine in Steam but rumble does not work (which works perfectly fine in 2.4G mode).

So unfortunately this is currently a pick your poison situation with this controller. The controller does have the option to bind keys or key combinations to its L4/R4 buttons but the configuration options are limited to the controllers buttons while Steam offers a plethora of more options (i.e. keyboard buttons which can be very helpful in some games). On the other hand, missing out on rumble also degrades my experience with a game.

I did a bit of research and apparently Steam added support for the controller a while ago. So this might be a bug, oversight, or limitation. In the case of the latter there might be little to do, if it's one of the former I'd like to kindly suggest the team to have another look at it.

Tested on Steam Deck and PC (Windows and Linux), the problem exists on all of them playing games on Steam. 8bitdo does has a software for Windows and Mac but it's only used for updating as far as I can tell. I have to admit I only tested on Windows briefly and to update the controller firmware as Linux is my main OS. So even if the 8bitdo software offers more functionality for this specific controller, I'd still like to suggest better support for Steam for a seamless experience, especially on the Steam Deck and for other Linux users.

Thank you kindly.