The Aurora has been built and has been flying for over ten years. It is officially a “No Comment” subject, just as the F117 was. However, there are pictures, the “string of pearls” and many radar trackings. It is one of the worse kept secrets of the skunk works.
Even if true, there is zero reason to suppose it more cost effective than a commercial space transport for such a mission. Government programs are not noted for their low procurement or operating costs.