CAPE CANAVERAL, Fla.—Two billionaires are putting everything on the line this month to ride their own rockets into space. It’s intended to be a flashy confidence boost for customers seeking their own short joyrides. The lucrative, high-stakes chase for space tourists will unfold on the fringes of space—55 miles to 66 miles up, pitting Virgin Galactic’s Richard Branson against the world’s richest man, Blue Origin’s Jeff Bezos. Branson is due to take off Sunday from New Mexico, launching with two pilots and three other employees aboard a rocket plane carried aloft by a double-fuselage aircraft. Bezos departs nine days later from West Texas, blasting off in a fully automated capsule with three guests: his brother, an 82-year-old female aviation pioneer who’s waited six decades for a shot at space and the winner of a $28 million charity auction. Branson’s flight will be longer, but Bezos’ will be higher. Branson’s craft …