> This is something I hadn't thought through before and made my morning bus ride substantially more interesting

Me too. :-)

There's no real rigour below, although there are hints about how one would grind out real predictions.

Let's start with writing down the Einstein Field Equation as G = T, where G is the Einstein tensor G_{\mu\nu} and T is the matter tensor, and where we use units that set c = G = 1, where that G is Newton's constant and with signature (-, +, +, +) and with spacetime indices ranging 0-3 and space-only indices ranging 1-3.

Next let's put ourselves into the Newtonian limit.

Let's treat M_{inertial} as T_{0}^{0} and identify the asymptotic behaviour of g_{00} (g being the metric tensor) as an energy and call that M_{gravitational}. M_i and M_g respectively for short.

There are several ways we can now break M_i = M_g, depending on how the breaking happens.

If M_i \neq M_g depends not on the composition of T but rather on position in spacetime, then people have already described modifications of G in a variety of theories that adjust different components of G (e.g. f(R) gravity). Most such models work very hard to suppress differences from General Relativity from the start of the matter era, rather than characterize the universe (or the solar system) if a difference were allowed to run. One would expect (in our Newtonian limit) to see differences in null geodesics in our solar system, with different results in the deflections of light from background stars as our orbit takes us to a position where the sun obscures their view. We would also expect different signal timing involving our various space probes as they move at various relative velocities at different and at different distances from the sun.

Where the modification of G depends on the composition of T then we need a change in spatial curvature g_{ab} (spatial indices 1-3) per unit M_i. We could do this with a bimetric (or multimetric) theory.

However, in our limit we can consider the behaviour of a mixed cloud of test particles such that neither its component nor the cloud as a whole significantly perturbs the metric.

Let's consider two classes of matter. "Heavy" matter has M_i < M_g. "Stubborn" matter has M_g > M_i.

Ordinary M_i = M_g matter has a "stubbornness" that is the "cost" of convincing it to move from one free-falling trajectory to another by the application of a force. It's "heaviness" is its tendency to follow a timelike geodesic wherever it leads in general curved spacetime.

Our mixed cloud has all three types of matter in it.

When the cloud passes near a massive object, it will separate. The "heavy" matter will spontaneously jump to geodesics that have a closer radar distance to the massive object, while the ordinary and stubborn matter will continue together.

Outside our test limit, we would prefer to say that all along the "heavy" matter was following different geodesics, i.e., we would drop the universal coupling of all matter to a single metric, and have lots of fun building a Lagrangian formulation.

Now let's crash our mixed cloud onto the surface of a planet that intercepts the cloud's free-falling trajectory directly.

Let's also employ an analogy between gravitation and electromagnetism. Muons and taus have more invariant mass than electrons, for the same charge. Stubborn matter has more M_g than ordinary matter for the same gravitational charge. Bremsstrahlung is different for electrons, muons and taus when their trajectories are altered in an electric field. Gravitational radiation will be different for ordinary matter and stubborn matter when deflected in a gravitational field. We would expect similar effects for other fundamental interactions for the stubborn matter vs normal matter. However, a stubborn matter test particle will have the same trajectory as a normal test particle, in the absence of interactions other than gravitational ones.

Continuing the analogy, a meteorite of mass m made of stubborn M_i > M_g matter falling onto a planet would penetrate more deeply into the planet's surface because it has more inertia to dump into the planet's matter through whatever interactions are available; muons and taus are more deeply penetrating than electrons because bremsstrahlung is the largest part of what decelerates them.

A meteorite of mass m made of heavy M_g > M_i matter will win a race to to the planet's surface with a normal matter meteorite, but its penetration should be the same.

I don't trust these intuitions in stronger gravity, but I'd fully expect compact objects with significant M_{i} \neq M_{g} components to be weird.

In fact, I wouldn't trust these intuitions at all. I'd really want to do an initial values formalism to see how heavy, stubborn and normal matter behave differently even in fairly trivial seeming scenarios. But that would take a very long bus ride!