The first postulate of SR is that an inertial observer sees all physics the same as he would if at rest. Including the effects of local gravity, to a high degree of approximation. This is Galilean relativity. If you stand still on the surface of an airless, rotating planet and throw up a ball, it will have the same tangential speed as you do and will rise and fall, from your point of view, just as if you were at rest. And it will come down and bonk you.
Now if you project the ball very hard, so that it soars high, then maybe tidal effects will have some effect, but that is very very small. Basically Galilean relativiy rules, and your theories of what happens are wrong.