As far as I can see, Fat Truck tours doesn't actually tour inside the National Park.
We've taken 2 of the tours on the school busses. It was fine. There were potty stops. There were also "get out and take pictures" stops (about every 1 1/2 to 2 hours). Yes, it's quite possible the windows get dirty as the roads are generally dirt/gravel packed. If it's raining the mud does get on the the windows. When we stopped the driver would "clean" the windows with a bucket and squeegee. He asked for helpers, as it would go faster that way. There were many who helped with this.