Any of the things that require you to use the gamepad instead of the Pro controller is going to require you to look at and/or touch the screen so there's no point in disconnecting the screen.
If you just don't want to keep charging it because it's on even when you use other controllers, you can open the panel on the back and unplug the battery and just use the Pro controller, but that means you can't change system settings, and obviously any games you're using need to support Pro controllers. Most do (at least most ones I've tried) but a few don't.