Momentum is part of the stress energy tensor, so it does matter. But momentum can be changed simply by looking at things from a different reference frame. Doing that changes the components of the curvature, but not any of the curvature invariants.

If you are just talking about a small body of constant mass as it gets closer to the speed of light in some asymptotically inertial frame e.g. due to an external pure force, that is one where the curvature invariants won’t change but the components will.

However, the more important point is that all of this depends on your frame of reference, since "approaches the speed of light" itself does. In the object's own rest frame, its speed is zero, and its inertia does not increase.

Heuristically, in the object's rest frame, if we assume that stresses inside the object are negligible, the object's invariant mass, which is equivalent to its energy in this frame, is the only contribution to gravity.

In a frame where the object is moving at close to the speed of light, again if we assume that stresses inside the object are negligible, the object's momentum does contribute to gravity--but its contribution is negative, to offset the increase in the object's energy in this frame (since its energy in this frame includes its kinetic energy as well as its rest energy). The net contribution to gravity (if we define "gravity" correctly as the invariants associated with spacetime curvature) ends up being the same. It's much the same as the way invariant mass stays the same in every frame because you subtract the contribution of momentum from the contribution of energy to get something that's invariant.

If an object is moving at nearly the speed of light relative to you, then you are moving at nearly the speed of light relative to it.

Do you think that your mass changes because an object moves at nearly the speed of light past you?