To me a TransAm means the TransAmerica route, not just any route across the US. So August may be a great time to start a trip across the US, but I doubt that it is a great time to start a TransAmerica if going east to west.
I still think west to east would be my choice, but Mr Bent has a point. It might be worth looking into whether a TransAmerica and Western Express combination would work. It sounds promising. AC says the Western Express "can be ridden from mid-May through October, depending on weather". Go farther south if necessary.