Which object in the solar system is essentially stationary relative to all the other objects in the solar system?

The correct answer is the Sun. If compared to the solar system, it is stationary and other planets circle around it. This is not completely true due to many facts like the sun revolving also and the center around which it rotates is not exactly in the middle of the Sun, but it can be treated as such.

Assuming we are speaking principally about our solar system the Sun remains at the center of the gravitational orbital plane for our planets. There force it is essentially the most stationary.