I am a Ph.D. physicist, and this is a relatively simple application of celestial mechanics.
However, anything more complicated than this will require an approximate solution, since there is no general solution to the three-body problem.
My plan is to determine the plane which the smaller body lies in, to determine the center of mass of the system, the total energy of the system, and the angular momentum, and use this to calculate the correct formula for the conic section of both objects.