

I know this doesn’t solve your immediate problem when using GNOME, but if you are desktop curious, KDE natively offers sharing a portion of the screen or even creating a whole new virtual monitor with its own resolution, wallpaper etc that you can share.
So my typical workflow is to have it create a 1080p virtual monitor above my real monitors, and share that, since sharing a full 4k monitor makes it hard for people to read. Then I can show whatever I want on that monitor without worrying about exposing something from my main monitors.
I’ve used a registry editor on Linux before to pull the Bluetooth pairing keys off my Windows partition so I could add them to my Linux Bluetooth config so the Bluetooth device would work on both systems.