How to disable "Mirror Displays / Extend Desktop" touch bar prompt?

I have a 2019 MacBook Pro (running Catalina) attached to the same external monitor as another computer. Every time I switch inputs on the monitor from the other computer back to the Mac, the application-specific buttons on the Mac's touch bar disappear and are replaced with a "Display Connected: [Mirror Displays] [Extend Desktop]" prompt, which I have to dismiss to get the application buttons back.

I switch back and forth multiple times a day and this prompt is pretty obnoxious and useless, interfering with any actual uses of the Touch Bar. Is there a way to disable it?