Does a perfectly rigid small object raise the Schwarzschild solution energy - a perturbation approach

Let us again look at the rubber sheet model of general relativity.

Let us have a small, flat, lightweight object which is relatively rigid.

We have a heavy weight embedded into the rubber sheet. It makes a pit to the sheet with its weight.

If we embed the small flat object to the sheet, will it tend to move closer to the heavy weight or farther away?

The flat object makes a "perturbation" to the system. It slightly deforms the geometry of the sheet.

Let the flat object initially be very far away from the heavy weight. The flat object is "relaxed" - its own deformation energy D_f = 0.

The rubber sheet shape close to the heavy weight has minimized the deformation energy D plus the potential energy V of the weight.

When we move the flat object closer, it tends to increase D + V, because it changes the shape of the system slightly away from the previous local minimum of D + V.

The deformation energy D_f of the flat object itself grows from zero to some small value.

Is it possible that moving the flat object closer could decrease the energy of the complete system? Yes, if the weight of the flat object is large enough. The potential energy V_f will decrease.

But if the flat object is lightweight, it will move away.

The next question is if a similar perturbation argument shows that a mass under the Schwarzshild solution tends to repel a lightweight, relatively stiff small object.