It is impossible to see any object that would be traveling that fast, thus your premise about such an object's distance must be wrong. Very high red-shift quasars have been found, from when the universe was less than a billion years old.
The Big Bang Theory is supported by extensive data. Six prominent facts are:
** The red shift of almost all galaxies, getting greater as their distance increases.
— This shows that the galaxies are flying away from each other, at greater speeds at greater distances.
** The cosmic microwave background radiation.
— This is a remnant of the radiation from the Big Bang, and has cooled over time to the exact temperature predicted.
** The variations in the cosmic microwave background radiation.
— These variations fit theoretical predictions, and were caused by quantum differences near the start of Big Bang.
** The proportions of the lightest elements and isotopes.
— This helps show that the calculations for nuclear interactions immediately following the Big Bang are correct.
** The changes in galaxies as we look further away (and thus back in time), with distant galaxies more primitive and having fewer heavy elements.
— This shows some of the changes in the universe since the Big Bang, and confirms the deep time of the universe.
** The change in the apparent speed of type 1a supernova as we look back in time, with distant supernova exploding more slowly.
— This shows that the light has been stretched out by the expansion of space over billions of years.