Q:

Which point in an orbiting body most closely follows its Keplerian trajectory?

Keplerian orbits can describe the motion of a negligible mass around a massive fixed central body, but they can also describe the motion of two bodies around their center of motion... with some constraints on their mass distributions.
But which point within each body is the point that traces out it's orbit? And which two points define the distances used to determine the center of motion? Are they the center of mass, or the center of gravity, or in fact a different point that needs its own definition?
Each body has a significant, non- negligible size and could have a non-uniform, or potentially a non-spherically symmetric density distribution.

A:

If one or both of the bodies have non-spherically symmetric density distribution, the orbits will no longer be keplerian. The gravitational force will not only be dependent on the distance to the central body (or barycentre in the case of two bodies with non-negligible mass) but also on the relative orientation of both bodies. 
For two spherically symmetric bodies, the c.o.m. (or c.o.g.) will trace out the orbit around the barycentre. The barycentre is determined by the same points.

