I don't have an 8bitdo Ultimate controller, but I have several of their other ones. They all work for me over bluetooth when I pair it with my PC and then use Steam to recognize the controller. They show up as a Switch Pro controller, meaning Steam won't be aware of the back buttons (I'm guessing you need to use 8bitdo's software to do that). But you can remap the regular buttons and the gyroscope in Steam's UI. If you use xinput mode, then you are limited to the controls on an Xbox controller, so no gyroscope.

That does mean you need to use Steam to launch all of your games, even if you bought them elsewhere, but Steam has a controller-friendly interface, so I prefer to do that anyway.

edit: Bah, I just saw @mp3's comment. Too bad that their Ultimate controller doesn't work like their other ones, that's a big step backwards.