Just move mouse cursor?

This script will store your current mouse position and then move it to the center of your desired screen. Set the screen you want to move to in the first line of the script. Make a copy of the script for each screen you want to move to.

When you run the script a second time, it will see if there's a stored mouse position and if there is it will return the mouse to the previous position. It doesn't matter which version of the script you use, if there's a stored position it will restore it. Running it again after a restore will move the cursor to the desired screen.

const targetScreenNumber = 2

if (globalState.csMouseMoveOrigPosition) {
    sf.mouse.setPosition({
        position: globalState.csMouseMoveOrigPosition
    })

    delete globalState.csMouseMoveOrigPosition

} else {

    const currentMousePosition = sf.mouse.getPosition().position;

    globalState.csMouseMoveOrigPosition = {
        x: currentMousePosition.x,
        y: currentMousePosition.y,
    }

    const targetScreenIndex = targetScreenNumber - 1;

    const targetScreen = sf.ui.screens.allScreens[targetScreenIndex];

    sf.mouse.setPosition({
        position: {
            x: targetScreen.frame.x + (targetScreen.frame.w / 2),
            y: targetScreen.frame.y + (targetScreen.frame.h / 2),
        }
    })
}

I'm with you on this!!! The fewer manual mouse movements the better!
I'll take a look shortly. :-)