Postulating the existence of a hypothetical negative mass, would the pseudotensor of its repulsive gravitational field have a negative stress energy? If so, why?
Relativists don't really talk about negative mass, they talk about energy conditions. If you were going to talk about negative mass, you would have to specify carefully whether you meant inertial mass, active gravitational mass, or passive gravitational mass. Stress-energy is a tensor, not a scalar, so again, it doesn't really make sense to talk about it as positive or negative; we can talk about whether or not it violates certain energy conditions. You can talk about the Newtonian limit, where only the 00 component of the stress-energy is nonnegligible, but in the Newtonian theory, the gravitational field of a positive mass has negative energy density.
Yes, it would, according to Gerard t'Hooft: "...it could have been that Newton’s constant G turned out to be negative. Gravity would be a repulsive force but that does not seem to contradict anything. Until you compute the energy of gravitational waves: that energy would have been negative, which would be somewhat disturbing from a physical point of view."