The thing is, that when I store in C++ my coordinates as float or double and calculate the fps by myself, I have a floating point precision time_step equivalent and can use this to update my coordinates as precise as possible. This will result in a much smoother movement, because I have a precise arithmetic, even when I am forced to transform the float XYZ back to var (but not using these values for my floating point coordinates, of course!).

I doubt that you see stuttering just because the entity snapped from 0.000 to 0.001. Stuttering comes from multiplying with the low-resolution'ness of the time_step factor.