As long as you're fine with importing AWT, you can use Robot anywhere, even if you're using LWJGL or NEWT for the other native mouse handling. Robot just moves the mouse and doesn't require a window to work within, or anything like that.
Ok but in a (near?) future, it will be necessary to remove any AWT dependency to benefit of an excellent cold startup time with the plugin 3, that is why we have to keep in mind other options.