You can... just move the mouse down to bottom of the screen. - very basic way to "hide" the mouse, somewhere that only 1 pixel of it remains. You can activate virtual mouse mode with LB/L1 and R3/RstickC and move the mouse that way if you are playing in a way that your mouse is not close.

My favorite thing to own for TV gaming is a wireless keyboard with embedded touchpad mouse. I am Ardyn you can trust me etc.