Can I turn display mirroring on and off with a shortcut?

Yes. With more than one display connected, press Command–Brightness Down to toggle macOS video mirroring. On many Apple keyboards, Brightness Down shares the F1 key. Use the key’s brightness function. Depending on your function-key settings, you may also need to hold Fn. This is a macOS keyboard shortcut and does not require a BetterDisplay shortcut assignment.

For separate start and stop actions with chosen displays, use BetterDisplay's actions in the macOS Shortcuts app. Add Mirror Screen and select the source and target displays for the start action. In a separate shortcut, add Turn Off Screen Mirroring and select the screen whose mirroring you want to end. See mirroring actions for their parameters.

The system toggle changes video mirroring. It does not start or stop BetterDisplay full-screen streaming or Picture in Picture. Those are separate presentation methods. See mirroring, streaming and PIP if you are choosing how to show a virtual desktop.