We present a new 3D magnetohydrostatic (MHS) direct elliptic solver for extrapolating the coronal magnetic field from photospheric boundary conditions in a manner consistent with an assumed plasma distribution. We use it to study the uniqueness of the reconstructed magnetic field as a function of how significant the plasma forcing is on the force balance of the magnetic field. To this end, we consider an analytic MHS model as ground truth. The model uses two free parameters to decompose the current into two parts: a magnetic-field aligned component and a cross-field component. We perform a comprehensive study of the 2D parameter space to understand under what conditions the ground truth can be reproduced uniquely. We find that current oriented perpendicular to the magnetic field has a smaller solution space than the same amount of current oriented parallel to the magnetic field, and so MHS regimes with larger proportions of plasma-related forcing may be a promising avenue towards finding unique magnetic field reconstructions.