Hello, when you stream via Discord, you need to start your stream and then return to the video you want to share... if it's on Prime, you should stay on your "Prime" page. Otherwise, your stream will be black.
Could you try this version of the client software to see if it improves things? https://discordapp.com/api/download/ptb?platform=osx
Please keep in mind that the link contains a preliminary version of the software, which has not yet been officially launched; so it may crash occasionally. If you encounter issues while using it, feel free to revert to the stable version of the client software!
Give it a try, and don't hesitate to let us know the results by replying to this message!