One reason why I said that Relativity is juvenile is the postulate that the speed of light is a constant. Michaelson and Morley got a null result for measuring the effect of the Earth's motion on the speed. The changes throughout the year were less than the uncertainty of the error in the measurements. So to a high degree of approximation the speed of light for the Earth is a constant. Raising this to the level of an axiom though goes beyond the supporting evidence. It works just as well as an approximation as an axiom. Einstein may have claimed a little too much but the Theory of Relativity does make a good working hypothesis.
For a circular Sun grazing orbit the orbital speed is about 440 km/s and fraction of the speed of light β = v/c = 0.0015. For a highly elliptical Sun grazing orbit the perihelion speed is about 620 km/s and β = 0.0021. So the elliptical orbit is the worse case for bodies orbiting the Sun and the resulting increase in mass is only about 1 part in 500,000. Relativity only has a minor effect on an orbiting body but errors that accumulate over centuries are noticible.