AFAIK Public Display only runs from the Time Race screen. So I always just run them from my main timing laptop. I setup my LED video processor to mirror part of the left side of the screen for the Public Display area. And then I shift all of other software windows
(ART, browsers, etc) towards the middle or right side of the screen -- so that the software windows don't show up on the LED panels.
I've never tried it, but I'm assuming that you can also use a Dual Monitor setup (e.g., Mini Display port to the 2nd monitor that only shows Public Display) and then HDMI to the LED video processor. So, basically, just have the LED video processor only mirror the 2nd monitor that only shows Public Display. Might be worth a try.