Q:

How to find instantaneous positions of a central body of specified mass that results in an osculating orbit?

If I understand correctly, osculating orbital elements such as those posted in JPL's Horizons represent the mathematical osculating (tangential or "kissing") Keplerian orbit about a specified location in space, without regard to what mass that body would need to have in order for that orbit to happen.
Instead, I'd like to calculate the position of a body of specified mass such that an orbit about it would have a specified state vector; at a given position $\mathbf{x}$ would have the velocity vector $\mathbf{v}$.
The goal is to formulate an alternative, more numerical, graphic, and slightly creative answer to the question What point does Earth actually orbit?
I'd like to calculate where to put the Sun with 1 solar mass such that it's orbit would include a given state vector.

Is this mathematically possible
Will this produce a different set of orbital elements than the traditional osculating elements?

A:

If I understand you correctly the state vector you would use would be defined in an arbitrary reference frame and you would like to find the position of the central body in that same arbitrary reference frame that would result in an orbit which includes that state vector. 
Mathematically there are infinitely many solutions to this problem, any position you choose for this central body will result in a keplerian orbit. Of course ruling out singularities such as zero velocity or zero position (with respect to the central body). 
Say you have some state vector with a small velocity. This may be the result of a circular orbit about the central body with a large semi-major axis and low orbital velocity. The same state vector may be the result of a highly elliptical orbit around a central body with the same mass, where the body is much closer than in the other case, but the spacecraft is near or at apogee. 
The Wikipedia page on Kepler orbits has a section on this initial value problem. The choice of location of a central body will just change the coordinate transformation you have to apply to the given state vector in the reference frame in which you also define the location of the central body and will simply result in different state vectors in the reference frame centered at the central body which will be used to determine the kepler orbit corresponding to this state.

