The spacetime geometry outside a static spherically symmetric object is described by an equation called the Schwarzschild metric. And as long as we are outside the body the geometry is independent of density of the body or indeed the density profile within the body. So, for example, if the Sun were to magically transform into a black hole of the same mass the orbit of the planets would not be affected¹.
Life is a little more complicated in your example because the Schwarzschild geometry describes a single isolated massive object, and you have two massive objects. The spacetime geometry of the two objects is not simply the sum of two Schwarzschild geometries because the spacetime curvature is a non-linear function of the mass. However, where the curvature is small (as it is in your example) it is a good approximation to just add the two Schwarzschild geometries and in that case the stability of the spaceship is unaffected by changes in density of the two bodies.
The same applies if the two bodies are rotating, except that the geometry is now described by the Kerr metric rather than the Schwarzschild metric.
As I mentioned above it is an approximation to just sum the curvature caused by the two bodies, and the question remains: if we did the calculation accurately would the spaceship at midpoint still be stable. I have to confess I don't know the answer. As far as I know there is no analytic solution for the two mass system so there isn't going to be a simple answer. However I can't see any obvious reason why the stability of the midpoint should be affected by the densities of the objects.
¹ not strictly true because the Sun is an oblate spheroid not a sphere and there would be small perturbations to the planetary orbits due to the Sun becoming a (near) spherical black hole. However this would be a very small effect.